What OpenAI Actually Did to Navier-Stokes

Can AI solve Navier-Stokes?

0:00But but in the Navier Stokes and that's why it's so interesting >> because the Navier Stokes has this nonlinearity but it also has this smoothing term. >> It has this counter force which is >> viscosity and so you're you're asking who's going to win and are they always going to win? So there are questions about credit commercial incentives accountability. Yes. But I do believe that the risks deserve real attention without being mired in those other things. >> Yeah. Those other things can be true and the risk can still be real. Okay, so just turn it off. And what does just turn it off actually mean?

Episode introduction

0:34>> Hello internet. This is your captain speaking Lester Nar joined as always by my co-host and our resident PhD Krishna Chowdery. We are about to get into what will be our most requested episode in the history of from first principles. Open AAI announced a solution to one of the millennium problems. Many of you have either seen that announcement or all of the fallout and drama as it relates to AI that has arisen since that announcement. And so, a quick overview of how we're going to structure this deep dive episode. We'll be lucky to get it under 3 hours. So, we're going to do

1:15something a little bit different than some of the mainstream coverage that starts with the drama. First, we're going to ground ourselves in the fundamental understanding of the mathematics as it relates to the OpenAI solution. What are the Navier Stokes equations and what is the existence and smoothness problem and how exactly did they solve it? And then we will obviously move into some of the drama and tragedy that has arisen as a result of this and other progression in AI over the summer into the early fall. This is going to be a nuanced episode. We're going to talk about the science and some of the larger complexities. If you want to understand science

1:56headlines, be sure to subscribe. As always, we are going to talk about the science from the ground up today because this is from first principles.

Navier-Stokes: Mathematics Meets AI

2:20It's not a question of if, but when. All of the people that have been paying attention to what AI has been doing in mathematics, that is what they are asking. When is AI going to solve one of the six open millennium problems? There's six of them that are left. They're kind of like the Infinity Stones. And I'd say AI is kind of like Thanos. You know, AI is coming. And the question was always when is he going to get the first stone? Well, it seems he might have got it. Okay. the Navier Stokes existence and smoothness problem might have been solved. And naturally, we were on vacation. I was in beautiful Lake Geneva in Wisconsin. Um, and somebody forwarded me a tweet. And this

3:02is what I was feeling. I made a meme about it, you know, where we're also anticipating GTA 6. Yes. >> And this is the ah, here we go again. Because honestly, I didn't I didn't want to keep talking about AI and math. We'd already done the Jacobian conjecture. we'd already done the remon hypothesis or a problem related to that both by anthropic but this this this is history if it's true um you know this is a big deal and it's not anthropic this time it's open AI and the announcement was met with a different kind of reaction than the other two in the other two there was a reaction of oh my gosh AI is so good at mathematics

3:43this is incredible there's a bit of an existential crisis in mathematics and so on and so forth. But here there was also something else. There was a fight. There were allegations of academic misconduct. Um there were questions about who deserved credit. There's questions about what Open AI proved themselves and what they maybe stole from other people's work. That's someone else being the NYU mathematician Tristan Buckmaster and anthropic researcher Levant Alpog who was the hero behind the Jacobian conjecture a few episodes back. So there's a lot to unpack but before we get into any of that as you said we're going to do something very deliberately.

4:23We're going to talk about the mathematics because at the end of this two-page document that Buckmaster published um he which really kicked off the controversy this this document where he talked about all the shenanigans that had happened behind the scenes. He makes the argument that if OpenAI has really solved Navier Stokes then they should just say so and he'd rather be discussing the mathematics than anything else. And honestly same for me same. So that's where we're going to begin. We're going to do the math first. We're going to do the drama second. So here's the road map for the episode. We're going to start from first principles and try to build up the Navier Stokes equations. They're set of two equations. And at the

5:03end of that segment, I want you to have an intuitive understanding of every single term in that equation. So it's not just a bunch of words and fancy symbols. At the end of that, we'll confront the actual millennium problem. This question of existence and smoothness. There is going to be some calculus involved. There's going to be some equations. But I have a lot of visuals, so hopefully it's not going to be that bad. And we'll describe it for our audio listeners, but I do suggest this is a very visual heavy episode, as most of our episodes are. Um, then we'll take a tour through the mathematics over the last several decades and get to the main event, which is OpenAI's 165page proof. Um, the claim is that they have

5:44solved it. Now, I am not a fluid dynamicist. Um and I'm certainly not an expert in the kinds of linear partial differential equations that the Navier Stokes problem is. This is a problem that has attracted some of the greatest mathematicians of our time including ter Terren Tao among others. So I'm approaching this from a perspective of a physicist who's trying to understand fluid mechanics. And so I went back to our trusty old Lando Lifchits course in theoretical physics um volume six fluid mechanics. Lando Lifchitz goes through some amazing amazing mathematics here and Navier Stokes is in the first in the early chapters because there's so much more to fluids than just Navier Stokes.

6:26Um shout out to my professor Robin Bruinsma who taught me this course. It was an amazing course where we had we got to apply some of my favorite mathematics i.e complex analysis to calculate the wing the the lift on the on a wing of an airplane using like a complex integral which was one of the most mind-blowing things that you could use complex mechanics to like talk about like real things you know um not just prime numbers as as we've been known. So, at the end of all that, we're going to be understanding the solution and then we'll talk about the drama and then all of the existential stuff about what all of this means because we're going to have to zoom out. This is not happening in a vacuum and there is a lot of crazy

7:09developments that are honestly happening happening daily. It it is really happening daily. And once Krishna has kind of walked us through the mathematics, I kind of want to look at the larger world that this announcement like happened in. Um there is legitimate disputes over the credit and how we think about acknowledgement in scientific discovery and understanding. Um and this is also happening in the context of an already existing and accelerating battle between multiple stakeholders as it relates to this issue. You have the Frontier Labs, you have the investor and venture capital community, you have politicians beginning to finally weigh in on this.

7:49Obviously, you have the general public who for many people don't really see real world impact on this in their everyday life. And you know for context around some of these AI tools I'm personally have been a user of them just as a technologist and someone who's interested in these subjects in the way that we cover on the show since GPT2 and I've seen what the progress looks like from a very direct personal context. And I do think that there is a challenge in discussing this issue right now because you have to fit into one of two boxes which is this is all hype and this is

8:31going to cause human extinction. >> Yeah. >> And in my view there's a lot of space in between this is all hype and this is going to cause human extinction. And I would love for us to be better able to have conversations within that gap. And so we're going to talk through some of the larger context to help frame this because this will be a technology, a transformation that will be with us for the foreseeable future. And I think it's going to be important to look at these two things and we'll look at some concrete examples of other things that have happened like the hugging face incident where there were some issues with OpenAI's models as it relates to the company hugging face because I think

9:12that will help really ground us around this shift from this is fancy autocomplete or this is just a better way to do search >> and this agentic world that has been actually a driver behind how Open AAI got to this solution. So, we're going to get into all of that. A real brief moment of housekeeping for those of you who may be joining us for the first time. Welcome to the best science show on the planet. You can find us on all platforms where you listen to your favorite podcasts as well as social for clips. Our website has all of the research papers, background material, and other ways to look into what's

9:52happening in Frontier Science Research at ffpod.com. Any engagement that you do on our videos on all platforms help us reach more people with what we think are some of the most fun and informative conversations about science. And we have some great merch. So with that, we're going to jump right in to what is the meat and bones of what's going to be, I think, a very fantastic episode. Yes. And we will start with the mathematics.

Building the Equations of Fluid Motion

10:20So these are the Navier Stokes equations. There's two of them. And our goal is to understand these equations. These equations are an equation of motion that describes fluids. How fluids behave. Um what are the forces that they exert? If I know what the fluid is doing right now, can I predict what the fluid will be doing at some later time? These are the equations that allow that to happen. And when I say they're an equation of motion, that's what I mean. They're like they're telling you how the the equation how the fluid is going to move around in some space. Now in classical systems the starting point for all equations of motion is always Newton's laws. Okay,

11:02Newton's first law is an object at rest stays at rest and an object in motion will keep going unless it's acted on by another force. So you know if you're in empty space and you just have a satellite that's moving in one direction, it's going to keep moving. What happens when there's a force? Well, when there's a force, there is an acceleration. It's not a velocity. It's not like the you need a force to keep it going. If it's going, it'll just keep going. If there's a force, it'll change how it's moving. So, that's called an acceleration. It's a change in velocity. And the relationship between the amount of force that I put on an object and how fast it's going to accelerate is

11:42dependent on the mass. It's a mass is a kind of inertia. It's like how much how much resistance to um a force do I have is literally what mass is. And there there's a pretty common demonstration that you see in physics classrooms in high school where you've got two rubber bands. They're the same rubber band, but one of them is attached to a block of wood and the the other one is attached to two of the same blocks of wood. And you can see that the the single block of wood accelerates a lot faster than the one with two blocks of wood. That's because the the two blocks of wood has twice the mass. So, it's going to have half the acceleration. There's Newton's second law. And Newton's third law is something about conservation of momentum, which is like, you know, if I'm pushing on an object, then that

12:23object pushes back on me. So those are the three laws. And if we want to understand how something moves around, all we have to do is figure out what is the nature of the forces on this object. And if we're good enough at the math, we can figure out how this object is going to move forever. This is partly what enables us, for example, to land on the moon, everything or at Mars, all of these kinds of obviously much more surface area than that, but this is why we can make predictions over such long periods of time. >> Yeah. Really, in order to get to to the moon, you just need Newton's laws. I guess you need electricity to like work the capsule or whatever, but like you know, if there was some analog version where like you had fluids that did

13:05computation or something, you could land on the moon with just Newton's laws. So let's take for example the mass on a spring. I want to go through a singular example of a very simple problem where I can have a new a mo um an equation of motion derived from Newton's laws and I can then figure out what that thing is going to do forever. Okay, a common example is a mass on a spring. Okay, a spring is a substance that obeys Hook's law. Meaning if I pull on the thing, it's going to resist and it's going to pull back. And how much it's going to pull back is dependent on how much I'm stretching it. If I stretch it twice as far, the pull back the force backwards is twice as much. That's called Hook's law. And you usually see it as um the

13:47force is equal to kx. K is how stiff the spring is. X is how far I've pulled. And so, you know, the the larger I pull, the more it pulls back. And the negative is because it's pulling in the opposite direction of how I'm distorting the spring. Okay. >> So, that's Hook's law. F= Kx. Now force according to Newton's second law is mass times acceleration. Yes. Right. And the acceleration is really a second derivative of the position of the x. Right? The acceleration is a change in velocity and the ch velocity is a change in position with respect to time. So it's really a double change which is why I have mass times a second derivative in

14:28position is equal to kx. Notice this is an equation of motion. I've got a single variable that I need to solve for which is X, the position. And the key thing is the force is dependent on the position, >> right? But the force is related to how fast the the position accelerates. So now I've got an equation that is in a single variable X and it's a differential equation with respect to time. This is something I can solve. Okay? And that's what most people do. You can solve this thing. It's a differential equation and if you solve it, you get a very nice solution. You get a very nice solution that is an

15:08oscillation. And I think we've got a little video of that. This is what happens. You take your equation of motion and you say, what is something where if I take a second derivative, I take a derivative once and I take the derivative again, I get the negative of the thing back. >> And that's cosine and s. The derivative of cosine is negative sign. The derivative of s is cosine. So I'd go from cosine to negative sign to negative cosine. Well, that's just the negative of what I started with. And there I have an oscilly motion. So this is the paradigm >> of how we do physics in general. We have Newton's laws. We analyze our system and we figure out what is the differential equation that the system obeys. And then

15:48we solve this. And if we're lucky enough, the differential equation is simple enough that the mathematicians have figured out how to solve this differential equation. And the physicist can then borrow that intuition and boom, you've got um a solution that tells you for any time I can plug in the time and I can tell you where the mass is going to be. >> And it it's a way in which we can now simulate and create replication such that we can predict how a system >> will act under certain circumstances. >> Yes. Exactly. And you can think you can make things even more complicated. For example, you can take the same spring and you can add friction. In this case, there's like some kind of damping force. And what that means is that in your

16:30differential equation, you've got a term that's related to velocity because like the faster I move through some like let's let's say I dunk this thing in honey, right? The faster I move through this thing, the more the honey is going to resist my motion. So that is dependent on the first derivative of of position, just the velocity. And if I add that term, then I get a new solution, which is the damped harmonic oscillator. And that's the red that you see over there. So in the undamped case, you've got an oscillation that just bounces over and over and over again. And then in the damped case, you've got an oscillation that dies out very quickly. For those of you who are listening to us in the car, um you can thank damped harmonic motion for your

17:10suspension because there's a particular value of the damping that makes all these oscillations go to zero very fast. It's called critical damping. And a lot of times car manufacturers want their springs, the suspension in their springs to have critically damped so that you know you get bounced on a on a what are those called? The holes, the potholes. You get you bounce on a pothole but you don't like bounce up and down forever. You just immediately bounce and then you go right back and there's like no bouncing whatsoever. And that's because the spring is critically damped. That has to do with how how much damping is happening. My point is you can make the equations of motion a bit more complicated and still be able to solve stuff. >> This is the visual I think of is seeing

17:51those meme videos of those giant trucks with giant suspension that's visible going over, you know, this humped dirt, you know, dirt track and they're chilling. >> Yeah. Yeah. Yeah. There's all these like Chinese propaganda videos, too, where they show like the luxury cars of China versus the luxury cars of Germany. And it's like the luxury cars of China are just like it's they're like on a cloud and then the ones like the BMWs like it's like okay well is that a new BMW or like you know you never know with those propaganda videos or is it just straight up AI. Anyways for every case the logic has been pretty simple. Okay the point is we start from Newton's laws. We understand what the forces are and then we try to set up the equations of motion. This works for springs. This

18:33works for planets. >> For a planet the force is proportional to 1 / r 2. how far away my thing is and it's pointing directly at the central the the central body and again that's a force that is a function of position and now I can solve for what the position of the planet is because I can set up a differential equation that does so okay I can't I can set up a differential equation for a three-body problem but crucially I cannot solve that so I can set up the equation of motion but I don't have a solution for it so there's caveats right okay so now let's try to understand fluids I see the setup we're going with here because we're saying we can understand the world around us and make predictions

19:14using math. >> Yeah. And Newton's laws. >> And Newton's laws for certain physical systems. >> Yes. >> And we just we just defined examples of those physical systems in which we can do so. >> Yes. >> Now I think where we're going is fluids maybe have some unique aspects. >> Yes. >> That make that different. >> Yes. It makes it different and difficult. >> Okay. >> It's different in the it's so so first of all it's same same a little bit because each part of the fluid still obeys Newton's laws. If I were to look at a single particle of of fluid, let's say I drop a dust grain into a moving river, right? That dust grain is going to experience forces because water

19:56molecules are pushing on the thing >> and those forces are going to make it accelerate. >> Okay? So Newton's law is still a work >> 100%. >> Right? The problem though is that Newton's laws are tenable and they're like tractable if it's a single thing that's like moving around in a fluid. For example, even in this cup, right, of of water, I have 10 the three molec No, no, no, it's not 10 the 3. That's only a thousand 10 the 23 molecules of water moving around. Okay. So, what do you expect me to apply Newton's laws to every single molecule that's in this water? It doesn't make any sense. That's sure that's one way of

20:36doing it. If I had infinite compute, which I don't, >> I'd like to I'd like to understand this thing at a level of granularity where I can make predictions, but maybe I don't have to worry about every single molecule and what it is doing. >> Well, you don't want to have to simulate everything because the system is so large as compared to what we were doing before. >> Yeah. And the system is large and it's self interacting is the other point, right? the water molecules are interacting with other water molecules and like it's just it's just a whole mess, right? So, it's not a new kind of physics. It's the same physics, but it's being applied everywhere in this system. >> Everywhere. All at once. >> And all at once. Exactly. And so, we need a new language. Okay.

21:17>> And perhaps here's the point. What do we want to actually describe about this fluid? We don't want to describe the individual molecules. As I just said, it would be crazy to try and describe the position of every single molecule as a function of time like we could with that mass on a spring. Right. Right. Um that

Velocity fields, divergence and incompressibility

21:34doesn't really make sense here. So instead of tracking individual molecules, we are going to track the velocity of the fluid itself. Okay. At different spots. And it turns out this has all of the necessary information that we need. >> Interesting. >> Okay. This is what we're doing. On the left hand side you see a video of sort of dust particles let's say that are in a chunk of a river. Okay. So we can see the dust particles that are moving from left to right. >> Now I don't want to worry about all of the dust particles. So instead what I worry about is what is the velocity of the particles at that point. And those can be represented by little arrows that

22:15just represent where the dust particles are going at that point. Now crucially the dust particles are moving but this velocity field that I've made of a bunch of arrows that are representing all of the velocities th that thing is stationary >> you see >> because it just represents this is the the direction in which I'm moving okay this is the central >> element >> that we are worried about >> okay >> okay this is a velocity field >> it's a field because it has a value at every point in space and time um and it's a velocity So it's got a vector. It's a vector field. Okay. And usually this thing is represented as a U. We don't want to use V for some reason. I

22:55think it's a historical reason. So in every case the U that we're going to see later on in this episode, that is a velocity field. It's a vector field everywhere in space. And it represents what the velocity is at that particular moment. So if I have a velocity vector that's pointed in this direction, that means water is moving past in this direction. If on the other hand it's over here then that means water is moving towards that arrow where the arrow is pointing >> and the component parts that make this velocity field are >> some position in 3D space which is our x y and z. So we know like where in this cube >> the velocity field that we're specifically referring to exists and then time allows us to create the arrow

23:36of time. Yeah. Of the direct like at that point >> at that point it's moving in this direction right yeah and it could change right maybe I I introduce a flow and then it starts moving in some other direction and so the velocity field can change. Um this is a very simple case right of just like particles water moving from left to right. There can be other more complicated cases. For example, this is the velocity field of a tornado. >> If I were to look like up the axis of a tornado, it's rotating around. And so imagine if you're looking like up through the eye of the tornado, you would see particles and clouds and everything moving >> around you, right? Like in a circle above you. And the velocity field for

24:17that looks like a bunch of arrows that are sort of circulating around. >> Mhm. >> Okay. >> Mhm. And the the color of these arrows represents how strong the velocity is. Okay? So the vector is big near the eye and as you move farther and farther away from the axis of the tornado, the vector field diminishes in magnitude cuz things are moving slower. >> This this makes sense. This is why when you have, you know, a tornado or a hurricane approaching you, if you live on the coast, the outer parts you you can feel it coming. >> Yeah. >> Because the outer parts are a little bit slower than the inner parts. >> Yeah. when you get to the inner parts then it's like okay then the palm tree is like >> on the side >> on the side right um and so this is the basic object in modern fluid mechanics

24:58it's not the velocity of water as though the entire ocean is a single baseball or entity that's just moving as one right it's no it's different parts of the fluid are moving in different ways and we can represent that using these little arrows that are in every single part of the fluid that's actually a really important mental picture for me to kind of reframe how to think about it Because my initial mental model is this is one discrete thing and we're trying to look at it as one discrete thing. >> Yeah. >> And what we're sort of saying is in order to get back to this idea of we want to be able to make predictions about a system. uh we need to look at it from this different vantage point which

25:38is uh velocity fields which help us look at uh points in space and the velocity at those points in space >> which then are also going to interact with each other over time. >> Exactly. Yeah. And so if we wanted to for example um calculate like what a dust particle is doing in that velocity field all we have to do is like follow the arrows right we can still recreate >> what we want which is like what like if I were to drop something what is the position of that thing right? All we have to do is follow the arrows. But the velocity field is something that holds all of the relevant information. That's the point. Yeah. And it becomes something that is a tenable object. It's something that we can understand and we don't have to worry about the 10 to the however many stuff that the fluid is

26:20made out of. >> It's less compute constrained for lack of a better way to put it as one way to put it. >> Yeah, that's certainly one way to put it. Yeah. And so before we start applying Newton's laws to this thing, let's try to understand this velocity field a little bit further. I just showed you two examples. The first one was water is moving from left to right. So all of the arrows are pointing from left to right or the tornado where the the clouds are moving in a a circle above you and so all of the arrows are pointing in kind of a circle, right? They're like circulating. Um in both of those we saw the particle picture and we were like, okay, that seems reasonable. Now I can imagine a fluid doing that, right? I can imagine the tornado and I can imagine stuff like this. Now are

27:01there velocity fields? Are there are there are there setups of arrows that we can make up that don't make any sense in the real world? >> Okay, the answer is yes. Let's look at these cases. Here are two cases. The first one on the left is a bunch of particles that are just like emanating from the center. >> The expansion of the universe. >> Yeah, it kind of looks like the expansion of the universe. Exactly. Right. It's like all of the all of the fluid is coming from somewhere in the center >> and then it's just like expanding out. >> Okay. How is that possible? Where is the fluid coming from? >> Yeah, >> right. That's not possible. >> Just like in empty space, it's just like there's like just stuff,

27:42>> right? Like what is the source? >> Right. Like if there was a hose, if if I saw a bunch of particles come from the bottom and then there was like a sprinkler head or something and then it moved. Then I'd be like, okay, but here I don't see that. It's just the things are just coming out. >> I understood, >> right? Doesn't make sense. >> Okay. >> Okay. First thing does make sense because somehow the fluid is being created in the center and it's just moving out, >> right? >> Okay. Second one on the right. All of the fluid is is moving into a single sheet. That yellow sheet, it's like a sheet of paper and the fluid is moving from the top into the sheet of paper and the fluid is moving from the bottom into the sheet of paper. How is that possible? Where is the fluid going? >> Yeah. Right. Right. Right. Cuz you would imag It should like go like it should

28:23continue. I mean, if I saw the if I saw it shoot out >> to the side, then I'd be like, "Okay, that's possible." But I'm not seeing >> of a wall or something. >> In that vector field, all of the arrows are pointing down and there's no point arrows pointing out. In the in the vector field on the left, all of the arrows are pointing out and there's nothing that's supplying the fluid to to have those arrows point out. And so just to be clear, we're sort of using these as sort of like an abstract theoretical or mental model to test the four corners of the sandbox of how to do this. >> Yeah. We're trying to we're trying to understand like we've created this concept of a velocity field. >> Yes. >> Now is any arbitrary vector field okay?

29:04>> Right. >> And the answer is no. Because on the left hand side that's an arbitrary vector field. Like if I were to create those arrows that are just emanating from the center, if that was an electric field around a positive charge, totally fine. >> But for fluids, that's not fine. >> It doesn't, right? It doesn't fluids. >> And so we can we can formalize this with mathematics. The property that encodes this is something called the divergence of an electric field. Okay. On the left hand side, we've got something that has a divergence of negative. Negative divergence. That means stuff is coming in into a singular point. Mhm. >> On the middle we've got a positive divergence >> like that I think like the collapsing of a black hole or just for my own >> visual 100%. On the on the middle that's

29:46a supernova type like the positive divergence stuff is moving out and on the right we've got a zero divergence. >> Okay. >> Okay. The only types of vector fields that are allowed for fluids are the ones with zero divergence. Okay. Because that means there is neither a sync nor a source. You can't create fluids out of nowhere. Mhm. >> So whatever fluid you have, it's got to come from somewhere and it's got to be going somewhere. >> Mhm. >> Okay. So that this is a key constraint. >> Mhm. >> For our vector field, >> for fluids specifically >> for for fluids specifically. For other things, it's totally fine. Like for example, if we were talking about the electric field on the left would be a negative charge, in the middle would be

30:28a positive charge. >> Fine. >> Fine. Totally fine. But for fluids, that's that can't be possible. >> So I think this is so an important point here. The idea is this concept of divergence uh applies differently based on the underlying subject matter that you're speaking to. And so as we've gone through this progression >> of testing uh vector fields, it only can work when the divergence is zero. >> Is zero. Yeah. Any nonzero divergence and it doesn't make sense. Another way of putting this is something called incompressibility. You cannot compress the fluid. Okay. Those are the fluids that we're talking about. And if we go to the next slide, those are our two Navier Stokes equations. You can already see that we've already got one of them.

31:08>> Okay. >> Okay. The second one is that the divergence of this fluid velocity field is zero. >> Yeah. Yeah. >> That's already the second Navier Stokes equation. >> Okay. >> So, we're already you would say halfway through >> and and can you remind me again, we have our uh I guess delta symbol and our U symbol equals zero. That's what you're referring to. >> That's the second one. The second one there that that's you would you would say the divergence of you is zero >> right which was that di diagram where we saw the center point everything moving in one direction >> everything moving in one direction stuff came in stuff is going out and everything is fine you're not creating or um destroying >> fluid >> out of nowhere yeah and um for engineers

31:50you might you might understand this with um the continuity equation like for example this is applied very directly if you've got a pipe that has a large um cross-section and you've got water coming in and then the pipe like narrows into a small cross-section. >> For our audio listeners, imagine a wine bottle on its side. >> Yes. Exactly. Wine bottle on the side, but its base has been cut off. Yes. >> And then now you've got water or wine coming in from the from the base. Right. As it exits the nozzle or as it exits the tip, it's going to be moving faster because all of that fluid has been jammed in. Right? You cannot create or destroy fluid. So the amount of fluid that's coming in has to equal the amount of fluid going out. But that means that if if I've got a larger cross-section,

32:32then in a single amount of time, I've got some amount of volume coming in. And with a smaller cross-section, that same amount of volume has to go out, which means the velocity has to increase. Right? This is the continuity equation in action. And a mathematically succinct way of putting it is that dou equals z. The divergence of the vector field is always zero. And it kind of makes intuitive sense for someone like me, which is when I look at the sort of the area of the cross-section and the wider part, there's just more space for stuff to move. And we have to move the same amount of volume through a smaller space. So it necessarily needs to accelerate right at that point in order to >> for the divergence of you to equal 0.

33:12>> Equal z. Yeah. For that for that fact to be true, you have to you have to account for this difference in velocity. Right. Right. Now crucially this is an independent thing from that first Navier Stokes equation which is going to be the topic for the next um for the next segment. >> Okay so >> okay this is an independent criteria >> okay >> if we didn't have this >> you could have different types of fluids >> you could have for example compressible fluids >> like air for example is kind of compressible >> in the sense that like if you heat it up it's going to expand and if you cool it down it's going to contract. Now that's not a divergence-free fluid. >> The Navier Stokes equations, these two have to do with fluids in the sense of

33:53like water that is really not compressible. The density is the same everywhere. You can't adjust the density. >> So the a key takeaway for one of the key takeaways for this is is the concept of this idea of incompressible. Yes. Which we've just defined from mathematical first principles. >> Yeah. But but it really it just means you can't you know you can't pack stuff in. The density is always constant, >> right? It's like one of those, have you seen those like stress thingies? >> Yeah. The little stress ball. >> Yeah. Where you like squeeze it and then the the the thing is going to like shoot out somewhere else. That's an incompressible fluid. Right. Because if I squish it, it's got to go somewhere else. >> Mhm. Mhm. Right. Okay. >> It's not like one of those foam things where you can just compress it. >> Mhm. Mhm. Right. Which is actually an

Acceleration and the convective term

34:34important distinction. Okay. So this is helpful because that's one half of what people say when they say uh the Navier Stokes equations >> and it's going to be the basis by which I'm guessing we're now going to build an understanding of the second >> of the second one. Right. We've we've sort of gotten in intuitive understanding of what types of velocity fields are allowed >> first of all. Right. and kind of what the velocity field represents, which is a bunch of these arrows that tells you how the velocity is at that moment in space and time >> in a fluid. >> In a fluid, right? In an incompressible fluid. >> In an in an incompressible fluid. >> Yeah. Okay. So now let's try to apply this velocity field to Newton's laws.

35:16Okay. Or the other way actually to we want to apply Newton's laws to this velocity field. Now that's easier said than done. Um the challenge now is how do we talk about Newton's laws which has to do with individual particles in the fluid right it's like it's like this water molecule is getting forces from other water molecules so it's moving around Newton's laws has to do with the stuff it doesn't have to do with the velocity field >> so we have to now >> put Newton's laws in the language that we've just invented >> and just to take a quick step back just for for for my brain part of like it's like you know >> if you have a bathtub with a little boat toy boat in it for a kid, right?

35:56>> We talked about Newton's laws in the con context of like the boat in the water. >> Yeah. >> And then when we talked about velocity fields, it's like the medium of the water. >> Yes. The water itself. >> It's the water itself. And these are like different. So it's like the discrete object versus the medium in which an object or many objects may exist in. And the lens in which we look at them is >> from different doorways. >> Yes. In in the first one, it could easily I could easily just be like, "What is the position of the boat?" Oh, it's like right there, >> right? >> What is the position of the water? What what do you mean? Which water? >> Yes, >> there's there's um an Avagadro's number of water molecules that we're talking about, right? So, instead, we have to reinvent this new language of the velocity field. And now, what we're

36:37going to do is build this step by step. We're going to apply Newton's second law, which is force equals mass time acceleration. We're going to apply that to our velocity field. The first thing we're going to do is try and describe what is the acceleration of a fluid particle. >> Okay. >> Okay. Like in this velocity field. >> Okay. >> Okay. >> Okay. >> And um we're going to we're going to use traffic as an analogy. >> If you live in LA, you uh are very much are intuitively familiar. >> Yes. And you're probably listening to this while you're while you're in traffic. >> Okay. So traffic is kind of a nice analogy for fluids because you've got these working um agents in traffic, i.e.

37:21the cars, right? The cars can be the individual fluid particles and the highway is like the receptacle and the entire flow of traffic is the fluid. >> Okay. The challenge now is that we want to have a description for how the individual cars accelerate or de accelerate if they break or if they press on the pedal based on the stuff that we can see from a news helicopter because the news helicopter is up there and it's seeing the velocity field of these cars, right? It's seeing like, oh, over there cars are moving fast. Over here the cars are moving slower. And like that's how they like if you're on the radio they tell you, "Oh, the 101 is backed up from Sepulva blah blah blah

38:02blah blah, right?" Um, so they're actually watching the the cars, but they're watching an aggregate mass of cars. It's not like they're following a single car and seeing how that how how stuff is happening. So what we want to do is describe the acceleration of a single car based on what we see from from a chopper. Right? Now, first let's think about how cars can accelerate. Cars can accelerate and de accelerate as a response to the traffic itself changing at a specific spot. For example, suppose you're in the chopper and you notice that everyone is moving at like 80 m hour, right? It's a nice day and everyone's moving at 80 mph and all of a sudden it starts raining.

38:43>> Uh I was going to say 80 m hour in LA. Never happens. Never happens. >> This is an unrealistic example. >> Yeah. Yeah. Uh that's totally fair. Anyways, that's what I'm stuck with in the visuals. So, we got 80 mph and then all of a sudden it starts raining. >> Okay. And within 10 seconds, everyone slows down to 60 mph. Obviously, the cars have de accelerated, right? They've used their brakes to slow down. This 20 mph reduction happened in 10 seconds, let's say. >> And so, the car is accelerated, de accelerated at -2 mph per second. They're slowing as a response to what the traffic itself is doing. The entire

39:25traffic slowed down and so we slowed down. It's kind of like seeing like, you know, in the in the traffic map of LA, everything was green >> and then rain started happening. Everything turned yellow. >> Well, that means all of the individual cars also de accelerated. So, that's one way that you can de accelerate. And this is called the oilarian perspective because it's asking how the field itself is changing as a function of time at a given spot. >> And this is represented by the derivative of the velocity field with respect to time. >> This is makes sense, right? It's a velocity field. If I take the derivative of velocity, I get acceleration. And there you go. This is an acceleration >> of my particles, right? In this case,

40:08the thing is slowing down. But you could easily imagine after the rain ends, the things go back to 80 mph and it's going to speed up again. This is represented by a derivative of my velocity field. >> And so the idea is is here's a way to describe the velocity field as this system unique and different from the discrete individual car in traffic. >> Yeah. Yeah. And we can use the description of the velocity field and how the entire field is changing in time to describe how an individual car is experiencing acceleration. That's the key right? >> So, good, we've got at least some of acceleration down, but that can't be the whole story. Okay, that cannot be the whole story. For example, let's take a look at this air foil. This is a

40:50airplane wing where the air is coming in from the left and it's going over the wing and under the wing and then to the right. Okay. Now, crucially on the on the lower right, you're seeing the velocity field representation of this airplane wing. Okay. So the arrows above the wing, the velocity is very high because the air is like getting pushed up and then there's high pressure. So it's like >> these are the red arrows. >> Yeah, those are the red arrows above the wing. So there's higher pressure above the wing, lower pressure below the wing. And so there's slower fluid below the wing. >> Mhm. And and just the length of these arrows is meaningful here in that there there's more >> Yeah. It's faster. >> Faster. Yeah.

41:30>> Um and Okay, I'm already seeing where this is going. Okay. So, so this is the on the left is your particles that are moving, >> right? Like our flow of traffic. >> Yeah. Our flow of traffic. And on the right hand side is the velocity field. >> Mhm. >> But if I were to take a derivative of that velocity field, if I were to ask how is this velocity field changing with respect to time, the answer would be zero. >> Right? Because this is a static velocity field. If you go back to that picture that I had with the tornado, even though the the the particles were circling around >> velocity field, >> the velocity field was static because in the in this point the velocity is this way and then as you it goes around it's

42:11like coming around, right? And so so what does that mean that the particles are not moving around? >> Mhm. >> Another way of putting it with the chopper idea is like you know you have you have let's say an accident over here. Okay. on the 101 a way behind way way way ahead of the accident the um the the speed is very high but as you approach the accident it's very low right but the accident isn't going anywhere >> because the police are very far away. It just happened and so the police haven't gotten there. So the accident is not going anywhere which means that ahead of the ahead of the accident the speed is very high. Before the accident the speed

42:52is very low and that's not changing. >> The traffic pattern is not changing. It'll still be red here, green here. If I come back 5 minutes later because of how slow the response is, it'll still be red here and green here. So the traffic pattern has not changed in time. And yet, I think you would be um remiss to say, yeah, the the cars are accelerating once you get past the the accident and you're done rubbernecking. Yeah. You're like, "Okay, I got to get on with my life. Like, let's go." Right. >> Okay. A and so I just want to make sure that I'm I'm following this here because what part of what we're trying to say is >> we're trying to go from the velocity field >> and then say can we describe an

43:34individual object within that velocity field >> and how it accelerates >> and how it accelerates based on the the mapping we see in the velocity field. Exactly. And part of what you're pointing out here is there are contexts like the air foil or the traffic accident example you just described where the velocity field equals zero. >> And so that means you can't >> the the derivative the velocity field is not changing. >> Sorry. Right. So the right the velocity field it's static. >> Yeah. It's not changing which means you can you in some cases you may not be able to accurately describe certain particles within that context because >> yeah using just the time derivative of the velocity field you have to figure

44:15out something else about the velocity field is doing this and let's just think about this I think we've got in the next one you know the highway slowdown let's say there's a let's say there's um >> next one >> in the next one exactly so let's say there's road work ahead right >> um a car is going to slow down. >> Mhm. >> Now, all of the cars near the road work are going to be going slow and the cars before the road work are going to be going fast. The road work doesn't change where it is. >> But what is making the car slow down? It's the fact that it is approaching the road work. There is a spatial difference >> in the velocity field. >> You see, there's the time difference that we talked about earlier where the the rain came in. So, everyone slowed

44:56down. People slowed down here. If there's only a patch of rain here and everyone else is fine, but this patch of rain, people slowed down. Right? So that's a single spot, stuff is changing. But here I am accelerating or de accelerating because I'm moving through the traffic pattern. Right? If I go from a red to a green, I'm going to accelerate in my map. >> If I go from a green to a red, I'm going to deacelerate. Even though where the green and red are are staying exactly the same, the fact that I'm moving through it is making me accelerate or de accelerate. >> And so I think part of what you're trying to get at here is that the time derivative of the velocity field is not

45:36sufficient to describe the acceleration of an individual object. >> Yeah, it's part of the story. >> It's part of But it seems like the distance >> is potentially the other ingredients in the pie at least based on the example that you just brought. >> Exactly. Yeah. And that's a big part of it. It's the spatial difference. It's it's how the velocity field changes as I move from one space to another. Now, >> in that case, what I showed you is like, okay, you might think, well, um, what if, you know, I took a time derivative before. What if I take a spatial derivative of the velocity field? So, that tells me how much the velocity is changing from one spot to another, right? If I move here and then I move like um 100 ft over there, >> if I've slowed down, that tells me how

46:18much I should slow down. M >> that's not everything. >> Okay. >> Okay. There is a secondary effect. >> Okay. >> And to to understand that, let's let's look at that scenario again where you've got road work ahead. >> Okay. >> But now I've got two different cars that are approaching. The first car approaches that road work at 100 m/s. >> Okay. >> Okay. And the second car approaches that road work at exactly half the speed, 50 m/s. I think you'll agree that the guy that is moving faster is going to have to break harder. >> Yeah. >> Right. Because he's going to he's going to come upon that road work way faster than the guy who's slowing, >> right? He's going to have to be like,

46:58"Oh, there's road work. >> You know, fines are doubled. I better slow down a lot." Yep. >> So, there's a secondary effect. It has to do not just with how fast the traffic pattern is changing but also how fast the individual car is moving. >> Mhm. >> Yes. >> Yes. Because so and it again trying to use the terminology again. We talked about the time derivative matters. We talked about uh the the spatial matters and also the individual objects acceler uh velocity velocity speed itself >> is the third uh item in our uh uh cookie

47:39recipe here. Like all of these three things are relevant >> in order to be able to go from >> looking at a velocity field and describing an individual object within that velocity field's acceleration. >> Acceleration. Exactly. And so for the second scenario, we have to take into account both the velocity and we have to take into account >> the change in velocity, >> right? >> Okay. The change in velocity as a function of distance is like how how far away is the road work, >> right? >> And my own velocity as I approach that also has something to do with how far how fast I need to break. >> So it's going to be a multiplication of the velocity field.

48:19>> Yeah. >> Times the spatial derivative. That's the under upside down triangle. It's called nambla. I call it dell. But it's u dodll time u. >> Okay. >> This is called a convective term. And this what this convective term is doing now, it's giving us in the example we just talked about here. It's giving us the 100 m/s, 80 m second, right? That okay. multiplied by the delu which is the spatial how fast how how how much time do I have before the road work I mean how much distance do I have >> before the road work gets there right how much time do I have is dependent on how fast I was getting there >> yeah yeah yeah okay and that's okay

49:01>> make sense >> yes got >> okay and both of these terms combined give us our acceleration >> we've got the oilarian perspective which is how is the traffic pattern itself changing for example >> is it going from green to red if rain came down or something or if there was an accident, right? Then at a specific spot that highway would go from green to red on our Google maps >> and this >> and that's the first term >> that that's our velocity the derivative of the velocity field >> with yeah with respect to time right the veloc the field itself is changing the time >> the color is changing on the on the Google maps right >> right on the right hand side the Google maps color is staying the same but I am moving through

49:41>> that portion of the yellow or the red >> and so I'm going to be breaking or accelerating based on that >> and a more and effectively a granular way. There's like a bird's eye and then there's a like a a a zoom. >> Yes. And both of those are matter. And that's why it's called actually the oilerian perspective because that's kind of the bird's eye view. And then the lrangeian perspective is like the the nitty-gritty what's happening there. And that's the conceptual trick. Okay. And >> in in in the Navier Stokes equations, we're writing Newton's laws for the person in the car for the car itself based on the data that's collected by the chopper. The chopper is only >> finding you. The chopper only knows the the green and the red and and everything

50:22like that. It's seeing these aggregate >> traffic patterns. But from those aggregate traffic patterns, I could tell you if a car is over there, how fast is it going to have to accelerate or decelerate? >> Mhm. Mhm. >> Does that make sense? >> Yeah. We're trying to um again describe the acceleration or deceleration of an individual thing in an incompressible fluid >> from a vantage point that is only the vector field. >> Yes. >> Which which um the point being the vector field in and of itself does not have enough to describe the acceleration of the individual object until you >> until you do all of this. Until you do

51:03all of this. >> Yeah. Uh, is that the right way to say that? >> I think that is the right way to say it. Yeah. Exactly. And so now we have an acceleration. Right. >> Right. And all we need for that other part of Newton's laws, right, is F= MA. We already have the A, the acceleration. What is the mass? Well, the mass is just for a fluid, it's like the density. >> Yeah. >> Right. It's just um I could do a F= ma per volume >> and then just say, okay, density multiplied by acceleration. That's my >> that part of Newton's law. >> That's my ma. >> Yeah, that's my ma. And that's exactly what we get. We've got the left hand side of Navier Stokes equations. Okay. The first Navier Stokes equation. >> Right. Because density multiplied by acceleration. >> Right. Right. Uh because our

51:43acceleration is everything in this in this larger bracket >> which is what we just described. >> Yeah. >> Um which is trying to describe the acceleration of an individual object from the helicopter. >> Exactly. >> Uh in Okay. Got it. So now again we're trying to go we're trying to use Newton's laws as a language to describe stuff that's happening in an incompressible fluid. >> Yes. >> Um and we've now have half of Newton's force time mass equals acceleration for an incompressible fluid with the description that we've just gone through. >> Exactly. And now I want to before we

Why Fluid Motion Is Nonlinear

52:18even start talking about the other side of that equation which is the force. I want to linger on this acceleration term a little bit more especially that convective term because that convective term you might have heard that um the Navier Stokes equations are nonlinear. >> Yes. >> Right. Compared to linear differential equations the Navier Stokes equation is a nonlinear differential equation. >> That second term the convective term there >> has everything to do with that. Okay. So let's linger on that for a little bit longer cuz I want to show you exactly what all the trouble is. And just to reiterate here, the convective term that we have on the right was when we were talking about the car in the yellow or in the red moving from the 100 meters per second to the 80 meters per second describing the acceleration in this

52:59slice of the larger >> system. Yeah. And exactly. And crucially, it had to deal with its own velocity and the spatial >> difference in the velocity. Right. That's why it's nonlinear because there's two there's a compounding effect that is happening. Okay, let's let's take a step back and let's look at for example Maxwell's equations in a vacuum. Okay, these are the equations of electricity and magnetism. They're described for example Gaus's law for electric fields is that um the divergence of the electric field is equal to how much charge there is. You know how before I was telling you that like fluids are divergence less the divergence is zero. But I showed you those two examples where the vector

53:40field was going out or the vector field was coming in. And that's because you can have negative charge and positive charge, right? That's Gaus's law for electricity. Okay? It just says that del E the divergence of the electric field equals to some charge. Now, crucially, the divergence of stuff is a linear operator. Meaning, if I've got one electric field and another electric field and I want to find the divergence of the two of the electric fields put together, I can just find the divergence of one and find the divergence of the other and I can add them up separately. So I've got if I want to combine two electric fields, I just do the math for one of them. I do the math for the other one and I combine them. And that's totally fine. This is why, for example,

54:21if I've got two charges, a negative charge and a positive charge, and I want to find the electric field everywhere, all I do is find the electric field due to one, the electric field due to the other, and I add it up. >> Okay? This is also why telecommunications works with electric fields, right? Like in this room, we've got Wi-Fi, we've got Bluetooth, we've got radio, we've got um our phones, we also have the light from the studio, right? All of these are electric fields, but they're all just moving through one another. They're they're like ghosts that are just like moving through one another. And if I can tune my radio to one thing or the other, I can listen to one station or the other. Or I can like my Bluetooth isn't

55:02randomly like interfering with the Wi-Fi. Like the Wi-Fi is coming from the router. There's a Bluetooth that's probably in my phone, but it's not like the Bluetooth signal and the Wi-Fi signal is like crashing. >> Mhm. >> Right. >> Mhm. >> Yeah. They they don't interfere with each other. >> Yeah. Yeah. Even like um in a radio, right, in in the car, like if I'm tuned to 91.5 KUSC classical radio, um the car is being inundated with all of the radio stations. There's also 89.9 coming at me and Kiss FM >> and also the AM, you know, talk radio. >> Not sponsored. We need our check. Yeah, these these are just some of my favorites that I listen to in LA. But like

55:43>> my point is the car is being inundated with all of these radio stations, but my radio can tune in on into a specific band. And it's not like the KISS FM radio signal is interfering with the KUSC signal is interfering with Rush Limba or I he's not. >> I think he's still on radio. >> Yeah, he's still on radio. >> We I think he's still on radio. He might be on YouTube now but or a satellite. But I think so part of what you're saying is like I don't have to account for the other radio stations. >> Exactly. >> It doesn't matter what And if I want to account for all I do is add it, >> right? If I want to know what is the electric field here, all I do is add up all of the contributions. It's totally fine. Mhm. >> Um in fluids we do not have that luxury. >> Okay. And that is because of that

56:25convective term. >> Mhm. >> That convective term is nonlinear. And this is what it does. If I add up two velocity fields that are separate suppose, you know, and before I was I was adding up E1 and E2, the electric fields. >> Yes, >> it's totally fine. >> Fine. >> Suppose I have two velocity fields, right? I got two hoses and I'm like crossing streams or something. U1 plus U2. Well, that's I shouldn't have, you know, whatever. like two two things of water uh and I and I and I add up, right? Like U1 and U2. >> Suppose I I go through the math with those. >> Okay. >> If it was linear, I would only get the terms in the green. >> Mhm. >> Right. Because I could just do the math for one U1 and I could do the math for U2 and I'd be fine.

57:06>> Mhm. You do the divergence for U1. >> Yeah. Divergence for U2 multiply by U2 and U1 and that'd be it. Okay. But because it's u multiplied by the divergence of itself, >> right? Or it's multiplied by the gradient I should say of itself. Not divergence because divergence has a dot. Anyways, the the point is I'm taking the the velocity field itself and I'm multiplying by the derivative of itself. >> Because I'm doing that, I have to do the distributive property. >> You have to do it twice. >> Yeah, I have to do it twice. And so I'm going to get these cross terms on the right in the red circle. >> Okay. And those cross terms are what make this whole thing nonlinear. And that gives us all of the richness of fluids. This is why fluids

57:48don't just go through one another. >> They crash. There's rapids, right? There's when when two rivers meet together. They they mix and they they they have turbulent flow. Mhm. >> You know um when I when I was in India last time um we visited D praag in the Himalayas which is the birthplace of the Ganga where like you you have like Bhagirati coming from one side and Alakananda coming from the other side. There are two rivers from the Himalayas that meet in this very holy place in the Himalayas that begins the Ganges river. Um and you could you could see the whirlpools forming like right at that spot. It was it was amazing. And and the the reason for the whirlpools and the reason for all of the chaos in the birthplace of the Ganga is because of

58:29those two terms on the right hand side. >> You know what I mean? >> Yeah. And so now you can already see that there is trouble with this equation, >> right? And I I just want to take a quick step back to make sure I'm internalizing what we're saying here. um when we talked about Maxwell's equations >> uh and we have you know a positive and a negative charge and we can we can just add them up. It's because part of what we're saying is is it's because we don't have that extra level of complexity when we talked about the convective term that's true for fluids. >> Yeah. Yeah. In the Maxwell's equation it's just d right. It's a single E. >> Right.

59:09>> Here there's two versions of the vector field. There's the vector field itself. Mhm. >> multiplied by the derivative of the vector field >> vector field. >> Right. And can we can we just briefly try to connect it back? >> I'm trying to connect it back to our yellow red traffic example just to make sure it lands in my brain because I'm I'm I'm having a hard time finding the leap from I I understood the convective term. >> Yeah. Um we have this idea that the traffic is red in one part, yellow in one part and we can look at it from the helicopter and we can look at the whole vector field. >> Yeah. And if a car is going from red to yellow or from red to green then it is going to accelerate even though red and

59:51green have stayed put. Mhm. And so would it like from that perspective, would it be like the equivalent of saying if that was it and that was the only level of complexity? It would be similar to Maxwell's equations and that if all I had to do was take a take a spatial derivative, >> right, >> of red and green and be like, oh, there's red here and green here and they're 100 m apart. So, you know, the difference in the velocity is divided by 100 m. That tells me the spatial derivative of my velocity and I'm fine. That's fine. But crucially, it also has to do with how fast or how slow I was going into the thing, >> right? If I'm transitioning from red to green and I'm moving at a snail's pace of like 5 mph and all of a sudden now I have to get up to highway speed, I got to floor it.

1:00:31>> But if in the red I was already going at 30 >> and then now I got to get to 60, I don't have to floor it as hard. >> So there's two contributions, right? And that's that's what's giving us this nonlinearity. Yes. >> Is because >> fact that there's two contributions that we have to account for >> as we look at trying to do the mathematics for this. And that's where we're getting this is where we're getting having to do it the we have the green but we have to have the addition of the red because of that second >> because because of the fact that yeah there's two of them. So I have to do these cross terms. >> Understood. When I combine velocity fields when I combine let's say a flow in this direction and a flow in this direction I can't they don't just ghost

1:01:11past one another >> because they're interacting. >> Yeah. Yep. Okay. >> Make sense? >> Yes. Tracking. >> Yeah. And so that's why the Navier Stokes are so hard. >> Mhm. >> That's one of the reasons. Okay. Um so now let's get back to our Navier Stokes equations. Finally, we can now start asking what is the force that is acting on the fluid. So far we figured out how to describe the acceleration >> and we've figured out how to describe the mass, >> right? Um yes, >> but in order to have an equation of motion, we have to describe the forces that cause this acceleration. And then if we write it out then perhaps we can solve it >> right in the same way that for the

1:01:51spring we could solve mass times the acceleration. In that case the acceleration was super easy. It's just the second derivative of the position and then the force was kx. I had an equation of motion that I could solve. In this case already the acceleration is is quite difficult. >> Right? But now let's ask what is the

Pressure, Euler and the Missing Physics

1:02:08force? One obvious thing is pressure. Pressure is force per unit area. you know, higher pressure means that there's more sort of forces acting on um acting on your container. Like, you know, if if if you've got a um if if you've got a high-pressured gas inside of a container, that gas is pushing out on the walls of the container compared to a low pressure gas, it's not pushing out as much. >> Just hanging out. >> Yeah. Exactly. Like when you suck on a straw, what you're really doing is creating low pressure in your mouth and then the high pressure of the atmosphere pushes down on the liquid and forces it up into your mouth. That's a difference in pressure that you're doing. So crucially, what matters here? What is

1:02:48the force >> on the fluid? >> It matters what the difference in pressure is, not what the absolute pressure is. >> Right? There's absolute high like a,000 um 100,000 lbs per square inch right here >> um in this room. But I don't feel it because there's no pressure gradient. There's no difference in pressure, right? So I don't feel a force. Now all of a sudden if um you know outside there was super low pressure for some reason and we had like a a leaf blower in here um there would be high pressure in the room. Low pressure outside and we'd feel a wind that's going out. In weather systems you see this a lot right? High pressure the wind goes away goes away from the high pressure towards the low

1:03:28pressure. So one of the forces that we will have to worry about is the gradient in the pressure. How the pressure changes from one space to another. And it's negative because it's going from high to low. So the force is always from high to low, >> right? Right. Because it's going from a high it's always flowing from a an environment of high pressure to an environment of low pressure. So you can always have the negative in front of it because it's always going to be that way. >> Yeah. It's always going to go from high to low pressure. And so this is a force, right? The whole point right now is we're trying to describe what are the forces on the fluid. In this case, the forces on that velocity field are from negative to from from high pressure to low pressure. You're going to have velocities flowing, right? It's like if you're at a beach and you know those

1:04:10little videos where the surfers create >> Oh yeah. the the surfing thing on the from a beach into the ocean. Yeah. And it it's it's like >> the the water pressure is very high and built up at the top and then it wants to flow into the ocean which is a relative area of lower pressure in that context. >> In that context actually you're the next one that I want to talk about is the external forces on my fluid and in that case that would be gravity. >> Okay. >> Right. because the the um like the lagoon or whatever is slightly above the ocean, right? And so when you create that little tiny thing, gravity is going to force the fluid down that little canyon that you've built. >> So my example was actually it was not

1:04:50perfectly at apt because I was not yet accounting for gravity. >> Yes. And the next thing that we're going to do is account for external forces on the fluid. >> Okay. >> Like gravity, right? So on the right hand side we've got forces. >> Yes. the difference in pressure is one such force that is going to you know like if I were to increase the pressure here and decrease the pressure here and I started out with some velocity field those velocities those vectors are going to get bigger >> because there's going to be more fluid going through and the fluid is going to be moving faster similarly if I introduce gravity now the velocity is going to get bigger so that's a kind of force it's causing an acceleration right

1:05:30and so here we've already got somewhat of a Navier Stokes equation >> okay Right. Okay. We've got um a familiar example is gravity, but honestly that G could be a lot of things. >> That G could be um the stirring force, >> right? Like if I put a spoon in a piece of in a in a glass and I stir the thing, that's an external force that's acting on my fluid. >> So the the force can be a lot of things. >> Understood. It's not necessarily just gravity in the context we just talked about it. Mhm. Yeah. We just talked about gravity in that context, but I mean forces on fluids can be a lot of things. It can be like the the turbine in an airplane >> that's putting a force on, >> right? Because the thing is it's

1:06:11stirring a fluid and making it shoot out in one direction. >> Yeah. Yeah. Yeah. >> Um so >> that is about as far as one of our favorite mathematicians on this podcast, Leonard Oiler got. And I think this is a good point to talk about some of the history behind the Navier Stokes equation because that last equation that we had is one of the one of the oiler equations of fluids. So, so just and just to quickly kind of recap, >> we've started we're trying to describe um >> the acceleration of an individual entity. >> Mhm. >> Within a vector field

1:06:52from only having the context of the vector field >> and we are trying to describe it in the language of Newton's second law. Mhm. >> Force equals mass time mass time acceleration. >> Yeah. >> And we've built up now for an incompressible fluid. >> Yeah. >> All the way up to where Oiler >> Mhm. >> arrived in in this journey of trying to do so. >> Yes. He didn't get there on his own though. >> Okay. Oh, okay. Oh, >> okay. We had no drama till the end, but >> and and there is some drama here. He he there's actually some really weird drama that is very pertinent to today's drama of OpenAI versus all these mathematicians. So I thought it was it was kind of interesting to talk about.

1:07:33So Leonard Oiler, you know, the great Leonard Oiler, we've talked about him a lot in the Remon Zeta episode. We talked about how Leonard Oiler invented the Zeta function that then Reman later extended to create um the Remon Zeta function and the Remon hypothesis. It's a very similar story here. Oiler created the Oiler equations and then Navier and Stokes extended those equations to create the Navier Stokes equations that we know today. And obviously you want to name things after the second person who invented everything because you can't name everything after Oiler. But even Oiler inventing that first equation that wasn't on his own. Okay. There is a lot of shenanigans that happened in 1940s

1:08:14>> 1940s Berlin between >> 1740s. >> Sorry. Yeah. Seven of course. Yeah. 1740s Berlin between Leonard Oiler. He was um working at the Royal Prussia Royal Prussian Academy of Sciences. This is the German Academy of Sciences. Okay. And if you go to Berlin, you can find a little plaque for where Leonard Oiler used to live. So the German Academy of Sciences announces a prize problem about one of the great practical questions in the n in the 1740s. How much resistance does a body experience as it moves through air or water? Okay, it's a very pressing pressing question because I think around the time is when cannons are being used

1:08:56in warfare. So, it's like, you know, we want to know how far a cannon can go. It's it's their version of the DoD um creating, you know, a prize competition to try to understand how to do better warfare. Um, a guy named Jean Leon Dalm. >> Very good. For those who are longtime listeners, this is >> I usually do this, but I like Dalbe. So, um, he submits an essay. It's about fluids. He did not solve the resistance problem completely, but he helped introduce something that is arguably maybe a lot more important, which is this mathematical language of a velocity

1:09:36field to describe a fluid. He's the guy who invented that concept of u of the velocity field. You know this all this stuff that we've been talking about this new language of maybe let's not worry about all of the particles. Let's worry about the velocity field of this thing. That was Dal Bear >> and Okay. So what's interesting is I want to make a quick zoomin note to note that the the prize problem that he set out to answer. >> Yeah. was about how much resistance does a body experience when it moves through air or water >> and I just want to be that's the specific question he was trying to answer >> in that journey he did not answer that

1:10:17question he did not answer the question but he invented he gave us the language he gave us the language for it right and his submission didn't win because the academy of sciences was like oh it's um it's it's too it's too theoretical he also showed that the that velocity field has to be diverted meaning that douals like the fact that it's incompressible and things like that. So he already established one of the two Navier so equations dam um he submits this thing the the academy decides that it's not going to give out the prize. Um no prize was awarded but here's the thing one of the people that was on the committee was Leonard Oiler. So Oiler saw an early submission of Dalbert's work and he's reading this

1:10:59thing going yo this velocity field >> is a kind of nice idea. He takes that idea and not long afterward he presents his own general theory of incompressible fluid motion and that's the equation that we see on the right hand side. So on the left hand side Dalambert has already invented >> the velocity field and he's shown that the velocity field has to have zero divergence meaning it's incompressible no sources or syncs. Leonard Oiler takes that concept and says well I can actually make a Newton's law version of that. M >> so very similar to remon zeta where the bassel problem existed beforehand right the one plus 1 over4 plus 1 over 9 plus 1 over 16 so on and so forth

1:11:40>> leonard oiler saw the bassel problem solved it and then presented the zeta function in this case he saw dalbert's velocity field and he's like wait this is a good idea and he just ran with it and created the derivatives which are huge like the the >> the the derivative term that acceleration is massive to create that material derivative and then also say that okay the forces on it are going to be the gradient and the pressure whatever external force >> and just say what you just spoke to that was the convective term that we talked about earlier. >> Yeah. Yeah. >> Um Yes. Okay. And this is so interesting because it's um it it's always funny how these things

1:12:24basically we have Dalam who started the >> who created the vision to think about it this way >> and then Oiler was like let me take Newton's >> framework >> Dalbear's vision >> and combine these two things >> yes and he gets an equation of motion right he gets the first equation of motion for a fluid this is for an invisid fluid as they say because these are >> as we'll find out frictionless fluids. >> Okay, now obviously that can't be everything. And Dalbert sees this um it's it's kind of like a mathematical twist that's so perfect it's got to be scripted. Dalambert looks at the original competition he's like wait a minute that it was about drag. If I

1:13:06apply Newton's I mean sorry if I apply Oilers's equation to the concept of drag and I ask what is the drag on a body? I can exactly solve Oilers's equations around let's say a ball moving through a fluid >> and there's no drag >> like in in in the next thing we'll see this is called Dalbert's paradox okay he uses Oilers's equation that's up there and he applies it to a ball a sphere that is moving through fluid >> according to Oiler's equations if you were to solve with all those boundary conditions the streamlines just go up and then they come back >> there's no difference in pressure >> from from the front to the back. And so there's no drag. But obviously, as we

1:13:46know from our FIFA World Cup episode where we talked about the drag on a soccer ball, there is a lot of drag on a soccer ball, right? >> Just as just to also note, this was Adidas's peer-reviewed study on their new balls for this year's World Cup, and they did incredible deep dive studies into the literal drag of how they designed. It's great episode. Definitely go watch it. >> Exactly. And what we can see already is in real fluids, the streamlines aren't just meeting up right after. >> Yeah. >> Right. There's a lot there's this like region where there's turbulence and all this weird stuff that's happening, >> chaos, >> right? >> Oilers's equation does not account for that. >> So basically what we're saying is Oilers's equations only accounts for

1:14:27what we see on the left, which is effectively >> uh a system that's not actually exists in real in real context. Exactly. The military can't use it. >> Exactly. Yeah. And so 20 years after the whole competition and everything, Dalbert Bert says, "This is a paradox. This is not everything. There's got to be more." And this is the part that Navier and Stokes >> help us repair. >> Okay? They're the ones that come in and

How Viscosity Changes Everything

1:14:50they start accounting for viscosity. And so to now kind of quickly recap again, we we are trying to build this understanding of what is the construction of the Navier Stokes equation. We understand DM Bear started with this idea that the divergence equals zero. >> Uh Oiler has come in and used the framework of Newton's second law of motion to try to apply it to fluids. Maybe cheating off of Denar's test >> maybe a little bit >> maybe >> um and added for the force side of the equation uh pressure >> uh >> and just an external force and an external >> gravity or stirring or whatever >> or whatever it might be. And he was

1:15:30like, I did it. Look at me. I'm so brilliant. >> Yeah. >> However, >> when applied in practice to a ball like we just described, it does not actually describe real world systems. And what's interesting is the it's funny because the original prize competition that Oiler was a judge on was about resistance and drag and then he came up with the solution without thinking >> without thinking about resistance and drag. Am I like missing something? >> Exactly. So, so what he ended up missing was another force that he hadn't accounted for. Remember, right now we have two forces. We've got the difference in pressure. Yes. >> And we've got just external forces, whatever they might be, stirring, gravity, whatever. There is a third

1:16:12force on that side of the equation that is missing. And that has to do with viscosity. >> Okay, >> that's the resolution to Dalen Bear's paradox. So, in Oiler's ideal fluid, there is no viscosity. This is called an invisid flow. Um the fluids can slide past one another. These particles can just slide past one another and they can slide along the surface of an object without any friction >> and the flow divides very smoothly in front of the object and then it rejoins at the end of the object. There's no wake, there's no net resistance, there's no draft. Beautiful mathematics. It's different than uh uh diver like it's different than um the problem we were

1:16:53saying where uh water uh like form a incompressibility, >> right? This is different. This is still incompressible, right? Because the flow is coming in and then it's going out. >> It's just that they when they interact with each other, it just rolls off. >> Yeah. They just they just roll off. Okay. Yeah. They're just rolling off one another. It's not especially useful. We haven't taken into account friction and we have to take into account something called viscosity. Viscosity is simply the measure of how thick, sticky, and resistant to flowing a liquid is. >> So, we're going from thick with two C's on the left here to thick with three C's on the right. >> Yes. >> Okay. >> Yeah. Vegetable oil, not that thick,

1:17:33right? You drop something, it immediately goes through. Uh motor oil, also not that thick, but slightly a little bit thicker. Um, honey on the other hand, you drop something and it takes forever >> to get all the way down because there's so much resistance to flow >> in honey. >> The things as they try to move past each other are make it harder. >> Exactly. Yeah. And so for low viscosity like thin liquids like like oils, um, the particles are moving past each other with very little friction. But for high viscosity, there's a lot of resistance to that movement. And there in lies the clue in how we can mathematically describe this because we need we need to have a mathematical language to describe what is viscosity. >> So let's go back to our traffic analogy.

1:18:14>> Okay. >> Okay. Let's consider a multi-laneed highway >> like this. So we've got a shoulder that's 0 miles hour obviously cuz you're on the shoulder you better be stationary. You don't want to be one of those guys that like tries to cut in line and then the the cops catch him. So let's look at let's look at what exchange of traffic from one lane to another will do because in normal traffic obviously cars are going to change lanes. >> So I've colorcoded these cars based on their lane and based on their velocity. >> Sorry. And what I'm realizing earlier you specifically showed us only one lane example. >> Yes. >> Which was fine. >> Yeah. >> At the time.

1:18:54>> At the time. But now we've got multiple lanes and now the flow is going to crisscross >> in between which is how real liquids okay >> how real liquids do it right now what I've done is in in the initial let's say it's a four-lane highway um the the lane that's fastest is going at like 80 mph and the lane that's slowest is going at 20 mph. So it goes 80 then 60 then 40 then 20 and then you've got the shoulder. four lanes and it's in increments of 20 mph and then things start shifting around. Okay, you actually wouldn't notice a change in the flow because let's take for example the lane that's moving at 60 mph. It's

1:19:37going to inherit some faster cars from the 80 mph lane and it's going to inherit some slower cars from the 40 mph lane and the things are going to average out. >> Mhm. >> Right. Um, and so if I were to look at a graph, and that's on the bottom there. If we look at a graph of lane on the x- axis and the speed on the y- axis, it's a line because it goes from 0, 20, 40, 60, 80, and later on it's still going to be a line. >> There's been no change. >> Meaning that the velocity field has not changed, right? It's still big up top, small down below. And the velocity field has not changed from one time to another. Meaning there's no force here. >> Okay? Even though there's some viscosity, there's no force.

1:20:18>> Right. Okay. Yes. >> Right. >> Yes. >> There's still like lane changing happening, but there's no force. There's no change in the flow of traffic. >> Mhm. >> Now, let's consider a different type of profile. A type of profile where the shoulder is again zero and I've got the two slowest lanes both at 20 mph. And then I've got a lane at 30 mph, but the faster lane is still going at 80. Imagine that. like the the carpool lane is still going at 80 and then the rest is like still very very slow and the carpool lane has like a barrier so there's nothing happen and then all of a sudden the barrier gets lifted. What's going to happen? Well, the carpool lane is going to slow down a lot but also the other lanes are going

1:21:00to speed up because stuff from the carpool lane is going to come in and you know sort of push that lane forward. This is a bumper car scenario. Okay, not real but but you get what I'm saying? Yeah, >> like like before we had this curvy profile which we see on the lower left where um again we we don't have a line because it's slow slow slow then really fast. >> Yes. >> And with subsequent time it's going to level out. >> It's going to level out >> because of the diffusion of cars. >> Here there is a force right because I've changed the velocity profile. If I change the velocity profile that's an acceleration which means there's a force

1:21:40lurking in here. Mhm. >> Right. There's a viscous force that is lurking in here. >> And this is what's it's what's interesting is um just going back to our previous example uh because basically the system was evenly distributed in the the initial state >> based on how the variables would end up mixing. >> You get you had no net effect, no force. >> No force. >> In our second example here, yeah, >> uh it was sort of uh a not evenly distributed system. Yeah. And so when you look over time and then there's the mixing. >> Yeah, there's a mixing. There's going to be a change. >> There's some diverg not diver. Let me not use the word divergence. There's some change. >> There's a change over time. >> Over time, >> which means that there is a force. And

1:22:21notice in the first example, there was a straight line >> which means I had a positive derivative, right? Like the slope is positive, but the second derivative is zero. There's no curvature. >> In the second version, there's a curvature, right? Which is why when I inherit the fast stuff and I inherit the slow stuff, the fast stuff is way faster than the slow stuff, which is why I'm speeding up. >> The curvature is the origin of the force. >> That's the key. >> Yes. >> And so that is the form that the viscosity is going to take mathematically. And if we go in the next slide, we'll actually see what the viscosity term looks like. >> It looks like uh dell squar. del square

1:23:02means a second derivative in some sense with respect to space and there's your viscosity the mu the the Greek letter there that's telling you how viscous it is like honey would have a very high value water would have a very low value and that's the force the forces again has to do with the velocity field itself >> this is always key we want to describe everything except for maybe the external force as a property of the velocity field itself >> of whatever the system as I use in a colloquial sense as a whole >> um is a material idea here which is why we have all these are all derivatives because we're looking at the system level >> yes exactly and like it kind of reminds

1:23:43me of like when we were talking about the springs right the force on the spring had to do with the position in this case the force on the spring has to do with the velocity field itself >> right >> yes >> and on the right hand side just for those who are engineers and physicists I want to show you a a similarity between the viscos velocity term and the heat equation where if you have the heat equation and you if you have a temperature profile as time moves forward everything gets smoothed out because of thermal diffusion. The same thing is happening with viscosity. Viscosity is smoothing out the velocity differences right and that's what we saw was happening with the traffic analogy. There was a lot of high speed here and low speed here and viscosity was

1:24:25smoothing it out and making the flow a bit smoother. >> Right? I'm starting to see the connection between the words incompressible and smoothness that I keep seeing as it relates to the whole Navier Stokes piece. >> Yeah, we're we're we're starting to see that. So, the viscous term becomes a smoothing operator in some sense, right? Um and now which feels very natural as a property of trying to measure a liquid because in this context it's the it's it's sort of this contiguous thingy when you look at it from the vector field perspective. >> Yeah. Exactly. And and we don't want that contiguous thingy to have like little pockets of really high fluid or

1:25:05sorry really high velocity because if there's a little pocket of really high velocity well that velocity is going to spread out. >> Right. And that's what viscosity does. >> Right. Right? That's why in honey, it's really hard to have a small pocket of really high-speed honey. On the other hand, for water, it's it's maybe a little bit easier to have a pocket of high-speed water, >> but then it just depends on what time time. Yeah. Anyway. >> Yeah. Yeah. >> And exactly. And so now we can solve Dal Bear's paradox because now we have um viscosity in mind. So we can apply this thing called the no slip condition. This is something that we've um looked at with hypersonics and we've looked at in the FIFA episode where if you've got a

1:25:46object that is moving through a fluid um very close to the object, the the fluid is not going to be moving at all. It's going to be stuck to the object and there's going to be a boundary layer where it's going to transition between being stuck to the object and moving with the rest of the fluid. And that boundary layer causes the drag >> because the object is dragging the fluid with it. >> Right? And this is how we solve Dal Bear's paradox because now if we look at for example a golf ball >> that is moving in air the fluid is sticking to the golf ball and so it's creating a wake behind it because the

1:26:26golf ball is moving through the fluid and as the fluid sticks to it the the the part that's right above the golf ball is going to be moving with that sticky layer but it's also going to be moving kind of with that. And so you get all of these dynamics happening because of viscosity. >> There's this transitionary space between the surface of the object that is moving through the fluid >> and then the rest of space >> and in proximity to the surface of the object there is a transition where the fluid around the object is going to move on some gradient from sticking to the object to being a part of the larger system. >> Exactly. Yeah. Exactly. And so that's how we get drag. And so this is the resolution and this is the contribution

1:27:07that Navier and Stokes were trying to capture. >> Um Claude Louise Navier in 1822 he presented an equation on viscous fluid motion where he added that viscous term but he was an engineer as much as a mathematician and um he tried to explain the internal friction by imagining molecular interactions inside a fluid. This was 1822 so everyone thought he was crazy because there was no such thing as atoms back then. I mean there was this Greek concept of an atom by democrat or something but you know no one took that seriously and it's one of these rare um occasions where a good outcome came from a derivation that's based on a questionable premise um now we know it's

1:27:47not questionable obviously and over the following decades we had Koshi Pson and others developed the continuum theory of stress and how stress affects bodies >> and in 1845 Stokes George Gabriel Stokes derived the viscous equations from a completely different direction from this idea of stress and how you can deform objects. Um, turns out it's the same form as Naviier and so now that's why we're called the Navier Stokes equations. They never collaborated by the way. Stokes was a teenager when Navier died. Um but their names are joined because you know different physical arguments but they arrived at these two equations. And now these are the two equations that have

1:28:28been such a headache for mathematicians and physicists all these years. >> And so as I look at now these two equations the one at the bottom which was where we started >> is this idea that uh the divergence always has to be zero for an incompressible fluid. >> Yeah. You can't make fluid and you can't destroy fluid. You can't have a fluid have emanate from a point in all directions and you can't have a fluid all converge on a point from all directions. >> That's what the bottom one is effectively saying. >> Yep. >> And then now the top one which was the more complex one is we're trying to replicate >> um uh force on the right equals match

1:29:09mass times acceleration on the left. in in this construction which was Newton's second law of motion. We're trying to apply it to incompressible fluids. >> Yes. >> Um and in doing so, we constructed the left side of the equation first where we have our mass represented by our volume. >> Yeah, that's the density. The dens the density, excuse me. Uh and then we then multiply that by everything in the larger brackets which is our acceleration which we have the first term being the uh time derivative of the velocity field. >> Yeah. >> Um >> that's like rain coming in and changing traffic patterns >> patterns. Um but we also have to add

1:29:50into that now the convective term which is how an individual car within that vector field is uh uh changing its velocity over with distance. >> Yes. >> Uh over time. >> Yeah. >> And so that's the left side. >> Yeah. >> And that that gives us mass times acceler acceleration. >> Yes. >> Loosely speaking. Maybe there's a few details not quite right. and then on the our right. So there's there's sort of three terms that have complexity on our mass times acceleration on the left. >> It's really two terms I would say. Okay. So so I I so fair within acceleration it's has two component parts. >> Well no well I'm I'm nitpicking here but

1:30:31like it's really two terms because it's density multiplied by dudt. >> Okay. >> And then it's density multiplied by that second term. You know what I mean? Like density is not its own thing. That's a very No, that that's an important distinction and that that's helpful. >> And then on our right side >> Mhm. >> where we're trying to define force, >> we started with the obvious thing which was pressure. >> Yeah. >> And it's negative cuz high pressure always moves to low pressure. >> Then we moved to uh we had we we added our F which is some general some general force. >> Yeah. Stirring, gravity, whatever you wanted to call it. >> And that's where Oiler >> ended up. >> Yeah. Yeah. and he stopped and he stopped

1:31:11>> and we were missing this concept of drag or resistance >> and ultimately both Navier and Stokes from different directions ended up on being able to define viscosity as the missing piece which is again talking about there's a boundary layer between the object in the system and then the rest of the medium external environment and there's some gradient delta there of change and so we need to account for the viscosity because that applies a force as we look at all of this is that now like an accurate description of what we're looking at >> that is the Navier Stokes equation and we have just built it up from first principles you know >> as as someone who did not do any type of

1:31:52mathematics it's a testament to your way of taking this story from start to finish >> yeah it kind of makes sense though right >> it makes total sense like when you understand the underlying systems I think the hardest part for I think the hardest part really is is getting familiar with the idea of a velocity field >> you know Once you have that and you've internalized, oh, it's like a traffic pattern, then all of the other stuff kind of falls together because you can like kind of, you know, figure it out. >> That was the Oilerian perspective was the velocity field is the way we described it earlier. >> Very nice. >> Dambert, dude. >> A little dam. >> Yeah. And so now we can start asking at this point, we've got the Navier Stokes equations. We know what every term

1:32:32means. >> Yes. So now we can start tackling how to solve them, >> right? And because and part of what we also define through all of this is this is an extremely complex problem. >> Yes. >> For all of the nuances of each of the underlying terms which also have their own sub >> uh issues, sub subcategory issues. >> Exactly. And so we've got an equation of motion. Now can we solve it? like can we can we find a general like I give you a velocity field now you tell me what the velocity field is later right what is the flow going to look like later >> and to just come back to the beginning of the episode the reason we want to do this is because with Newton's uh

1:33:13equations that we were talking about part of it's like then you can use it to make predictions >> exactly yeah >> and that's how we do the space stuff and blah blah blah so what we're trying to do is take this equation of motion >> so that we can like what is the who cares and the answer is so we can make >> so we can make predictions We can give you one input, the velocity field, and then you can give me the output consistently that's true every time. >> Yeah, that'd be dope. >> That'd be great.

Solving Equations vs. Simulating Fluids

1:33:36>> Yeah. Okay. So, we have a set of differential equations, and we'd like to have solutions to those differential equations. Now, what do we mean by that? Like, in a normal algebra problem, if I write something like x + 2 = 5, I know what x is. X is three, right? I can like do the thing. In a differential equation, what we're asking is something larger. What is the function that solves the differential equation? In our case, it would be what is the velocity field that solves this differential equation. Let's take a simple example for let's take for example um the derivative of y in terms of t is equal to some r * y. Meaning um I've got some quantity. The

1:34:16change of that quantity with respect to time is proportional to the quantity itself multiplied by let's say some rate. That is an exponential, right? The the solution to that is e to the power of RT. And this is how people get rich. This is uh this is compound interest, right? You start with some amount of money and then the rate tells you the interest over time. And the more money you have, the faster your money grows. And this is a this is a very simple differential equation that we can solve very accurately. Right? This is a toy equation and the answer is very nice. Um, Navier Stokes is less accommodating. Okay. The unknown

1:34:57is not how much money you have like a single one-dimensional variable. It's a three-dimensional velocity field, right? >> So, it's a number at every point in 3D space, but that number has a direction, right? It's a it's a length and it's got a direction that's also three-dimensional. So it's it's it's quite it's quite massive like this this equation >> make it's and it's so funny because now I that I understand that it makes my brain hurt trying to say we want to solve for this now given that level of complexity which is now very clear what that level of complexity is >> exactly like the the the fluid is interacting with itself and everything. So this velocity field has to solve for

1:35:37that entire big thing, right? We got to find all of the arrows everywhere, >> right? >> In some sense. Yes. Right. For all time cuz the arrows might be changing >> over time >> over time. And we need to we need to have an expression for all of that. Now there are certain special situations in which the Navier Stokes is actually exactly solvable. >> Okay, it's not true that the Navier Stokes is just unsolvable. You can't solve this equation. There are situations, there are certain boundary conditions where you can exactly solve the Navier Stokes equations. Um, let's go through one of them. It's called plain coet coete flow. I never know how to pronounce it. >> I think it might be coet. >> Is it coete?

1:36:18>> Let us know in the comments. >> Yeah. In any case, French. >> Yeah, it it's probably French, which is why I think it's coete. Um, here's what's happening. You've got two plates. Okay, those are the Navier Stokes equations that we're trying to solve. Yes, >> I've got two plates. One plate, let's say, is the the table >> um completely stationary. Another infinite plate is moving in the x direction in just one direction at a certain velocity. And in between two plates, you've got water. You've got some kind of fluid. Okay. What is the the velocity profile of the water in between these two plates? It's very simple, right? There's a stationary plate. There's a plate on top that is moving at a spec fixed velocity in one

1:36:59direction. and there's fluid in the middle. So, what is the is the velocity field going through? Now, we know from the no slip condition that the fluid that's right next to the table that's stationary is going to be stationary and the fluid that's on the plate that's moving is going to be moving with the plate. What happens in the middle? >> It's actually pretty simple. >> You solve for the Nav Stokes equations >> and you get this profile >> right >> near the bottom where it's stationary the velocity is zero. And as you approach the plate in the top, the velocity gradually increases linearly. And so it's just a function of the height, right? The higher you are, the closer you are to the top plate, the

1:37:39closer you are to the velocity. It's a linear relationship. Boom. >> Exactly solvable. >> That that makes total sense. And again, we've sort of created an exactly solvable simple >> Yeah. >> Uh use case because, you know, by having the bottom plate be stationary. >> Yeah. having the bottom even if the bottom plate is actually moving in the other direction totally fine as long as it's the same like you know the same velocity and there's so much symmetry here >> they're only moving it on one plane >> exactly it's only one coordinate that we have to worry about so a bunch of the derivatives over there just go to zero like there's so much symmetry and geometry here that it murders all of the difficult terms of that equation and

1:38:19you've got this steady profile and you're good to go. It's a recurring theme in fluid mechanics which is if you've got a nice symmetric distribution of stuff you can solve things >> fine you're fine. Yeah. This is called coete flow. You can also do pipe flows like there's a handful of vortices and jets where you can obtain beautiful closed form solutions based on the symmetry of the situation. And just to ask as a a brief side question, part of the other part of why those are so solvable is because we have such a good understanding of geometry as like like these other mathematical disciplines, we can apply some of those learnings and concepts in a way that make these easy or it's not it's not not relevant at

1:39:00all. >> Well, no, certainly I think I think I mean for that one for that one it's like pretty easy because you can just take derivatives and they go to zero and whatever, right? But there's other ones like there's certain vortices and jets where you can manipulate the symmetry to cancel out terms and things like that. You can you can use our knowledge of symmetry, >> right, >> to to to make like transformations that make these equations much easier to handle. The the subtle note I'm just trying to make here is is these things don't happen in a vacuum. Like our general mathematical understanding% >> has applications in subtle small ways all through. >> All through all through >> Yeah. all through. And so for that's that's for a very nice symmetric thing.

1:39:40Formula 1 cars not very symmetric. I mean there's some symmetry I guess left and right, right? So sure you can see that it's symmetric that like you can flip the the profile, right? But a Formula 1 car is insane. The geometry is extremely complicated. The wheels rotate. The ground moves relative to the car. There's thin boundary layers on the bodywork entirely. Air is accelerated beneath the floor. The vortices are deliberately generated around these waves to cause downforce. Um the nonlinear term does not vanish in the other one. The nonlinear term actually vanishes >> just because of the geometry of the thing here. This is the entire problem, right? There's no known formula that you

1:40:22can write out for what a Formula 1 car the the velocity and pressure at every point around a Formula 1 car is going to use. So instead what engineers do even in this visualization over here what they're doing is using a computer to computationally solve the Navier Stokes equation around this boundary. Okay. And this is called computational fluid dynamics. If there's any aerospace engineers or hydrodnamic engineers in the audience, leave one in the comments and tell us your favorite uh computational fluid dynamics program. There's a bunch of programs out there. This is one that I just picked up like on GitHub. And they always flaunt the

1:41:02number of cells that they can um break down their space into because you know imagine a Formula 1 car in a wind tunnel, right? That's some like volume. >> In order to solve these differential equations, you have to break up that volume into tiny little grid boxes. And then you can solve the Navier Stokes equations for each of the grid boxes. And you just can put them all together. >> Yeah. You can be like, "Oh, the velocity is here, but the velocity next to it is something." The the velocity to the right of and left of it is something. So, I can calculate a derivative that way, right? I can be like, "Oh, the difference divided by the grid spacing is my spatial derivative." >> Yeah. >> Right. And and I can I can do this computationally one at a time, and the

1:41:42more cells I have, the the better granularity and the more accuracy I have to the real thing. This this makes me think back to our hypersonics episode where we effectively discussed the same thing. >> Yeah. >> And and then sort of in addition to all of the challenges we've talked about >> uh when you start getting going from supersonic to hypersonic things get even weirder. >> Yeah. >> And we sort of discussed why it gets weirder and how we try to deal with that with these approximations that we've sort of discussed here. Again, Navier Stokes is not unsolved in all cases. We found clever ways. >> Yeah. To like approximate

1:42:23>> Yeah. a solution, right? And for engineers, this is good enough. The point they're like, honestly, like Yeah. >> Open AAI didn't solve anything. We've been modeling a Nav Stokes since you were before you were born. >> Yeah. For the Ferrari engineers, it's not good enough because apparently they still can't do it. Like, guys, figure it out. You have Lewis Hamilton and Charlotte Clair. Like guys, come please. I've been waiting for so long. Anyways, the result is not a closed form solution. Okay, it's this enormous table of numbers that tells you what the what the velocity is at every given point. And you can put any arbitrary shape into it. This is like a meme now in aerodynamics for some reason. Like all these aerospace

1:43:03engineers like put a cow, a CAD model of a cow into their CFD simulations and they're like, "H how aerodynamic is a cow?" I've seen so many versions of this meme for some reason. I don't know why, but a cow is a very non-erodynamic thing that people are trying to figure out, you know, what is the lift on a cow as it goes through a wind tunnel. In any case, the idea is any arbitrary shape, you can put it into a simulation >> and you can turn the simulation one after the other and computationally solve the Navier Stokes equations. >> For all practical purposes, that's totally fine. It's an engineering solution. >> It's not a physics understanding.

1:43:45>> Yeah. >> Andor mathematical understanding. And it's it's fine that it's an applied phys. >> It's a spectacular practical achievement. I mean, this is how we get >> airplanes. This is how we get fighter jets. This is how we get turbines. This is how we get wind uh wind farms for like power. Mhm. >> Um it's but it's not the same thing as the exact equations and what do those equations look like, right? What what is the solution to this thing look like for any arbitrary boundary condition? >> Um a simulation has a finite grid, right? It advances in finite time steps. It rounds numbers at some point. And so

1:44:26the problem is engineers are pretty happy taking Navier Stokes equations putting it into a CAD like putting CAD model in there using computational fluid dynamics they are good to go they'll tell you what the drag is they'll tell you what the lift is what the down force is on a formula 1 car and so on and so forth >> and unless you're making who hypersonics it's fine >> it's fine mathematicians on the other hand they're asking well what about like the existential problem of like I've got differential equations. Um, are the

Can a Smooth Fluid Blow Up?

1:44:59solutions always well behaved? >> Okay, that's the question. Here's what I mean by well- behaved. Let me give you an example of why they shouldn't be well behaved. It really has to do with these two terms that are in complete competition to one another. >> And let me see if I can get this right. We're talking about the convective term on the left >> which accounts for our acceleration and we're talking about the viscosity term on the right which is a part of our force. >> Yes. And these two things do very different qualitative transformations to our velocity field. The viscosity term as I told you smooths things out. Right? If

1:45:40there's a tiny bit where there's a lot of velocity and it's surrounded by not that much velocity then it's going to smooth out. the velocity is going to leak through and you're going to smooth out the velocity function. The convective term on the other hand, what it does is transfer energy from large systems to smaller and smaller systems. So the convective term is making smaller and smaller structures while the viscosity term is making those smaller structures smooth out. M there's a constant competition between these two terms in the equation >> because depending on the size of the

1:46:20structure you're referring to the viscosity will be different and the convective term will be different and it's different across every scale >> yes and the question is does one win out over the other specifically does the convective term ever win out over the viscosity term can I get smaller and smaller structure to such an extent where I can have theoretically infinite velocity somewhere because the convective term is just creating smaller and smaller structure. Now, I want to be a bit more concrete here when I say the convective term is creating smaller and smaller structure. Okay, so we're going to talk about um this competition in a very >> real sense with something called a tailor green vortex. This is a standard test case that is used in computational

1:47:01fluid dynamics because what you can do is actually like solve this thing analytically and then you can solve it in a computer and compare the two. Okay. >> And then that'll tell you how good your computer simulation is because you have a ground truth that's like a functional mathematical form. Here's what the tailor green vortex looks like. This is the initial condition condition. On the right hand side, you're seeing the velocity field at time t equals z. So this is what you're starting out with. It's a bunch of ss and cosiness for the x velocity. It's a bunch of ss and cosiness for the y velocity. And it's zero on the z velocity. Meaning all of the particles are are only moving in the xy plane. No one's going up and down.

1:47:42Everything is moving side to side. >> Flat land. >> Flat land. In flat land, right? But there's bunches of flat land. Like it can still be a 3D box, but like the the stuff at the top of the box is still going side to side. And the the stuff at the bottom of the box is still going side to side. That's how we start out with. And crucially, all of these terms have signs and cosiness. So what that what that amounts to is you have vortices. >> Okay? you have a the signs and cosiness always means circles. Okay, if you if you if you come out with one thing from this podcast, signs and cosiness mean circles. But in this case, what's happening is you've got these vortices that are circular. Okay? And so all of the particles are moving around in circles like this. Like the ones up top

1:48:22are moving like this. Down here, they're moving like this. Down here, they're moving like this. Imagine a 3D box. And now I press play. >> Mhm. >> On this thing, >> what is going to happen? Well, let's see what happens. Um, we've got a bunch of math here that I'm going to very quickly go through. Okay, for those who are not watching, my my eyes just expanded to a very large size. >> Yeah, but it's it's not that bad. And and I want I I'll tell you what to focus on. >> Okay. >> On the left hand side is our initial conditions. These are our ss and cosiness. It's just sin x and cosine cosine y sin cosine z. So this tells you

1:49:03that like the the size of my vortex is 2 pi in size. Okay. Okay. >> Now I apply the convective term to these equations. Right. These equations tell me what the velocity field is. I can find the gradient of the velocity field and I can multiply it by itself. And then I'll leave this as an exercise to the reader or in this case the podcast listener as all of the famous science and math textbooks do. Um, but you can go through and do the derivatives and then you'll have to remember your double angle identities and trig. But what ends up happening is >> those derivatives are going to create terms with a sin 2x and a cosine 2x.

1:49:44That's going to percolate into the into the pressure terms. Mhm. >> You're going to get sin 2x and cosine 2x. And one thing that you should remember is sinx is a wave. >> Mhm. >> Sin 2x is that same wave squished. >> Yeah. >> You're creating smaller and smaller waves. The wavelength is getting smaller. >> The oscillation >> is getting squished. Okay. Now, what does that do to our structure of the vortex? What happens is remember before I told you that yeah that's that's the so remember before I told you that the Z component of my velocities was zero everything was

1:50:25happening yep in side to side right but now because of that convective term and because of those higher wavelengths sorry I should say smaller wavelengths there's going to be picked up Z velocities there's going to be interaction that are happening in the Z direction. You're going to push up stuff and you're going to push down stuff. And the way you pushing up push up and down, those structures are going to be half the size of your original vortex >> of and this is direct coming from the squishing. >> Mhm. It's coming from the squishing and the fact that the squishing is forcing stuff into the third dimension. >> So on the left hand side you see these vortices. Those blobs sort of represent

1:51:07where the vortices are. You can imagine the orange blob is a vortex going in one direction. blob is a vortex going in the other direction, but all of those vortices are X and Y. >> Okay, they're they're revolving in this way. >> And they sort of have sort of, for lack of a better term, a consistent nondisturbed structure for time t equals z. Like that's what we started with. And when we press play, all of a sudden, you start getting these vertices. You're getting these interactions between the vertices up and down. >> Mhm. And the interactions those structures that are created in that interaction have a smaller and smaller size >> because of that convective term.

1:51:47>> Okay. >> Okay. >> I see. I see. >> And if we continue this at infinitum, it's going to get even smaller >> and even smaller and even smaller to at some point you would think you would think, right, that like it's just going to get smaller and smaller to infinity. You're going to get smaller and infinite decimally smaller structures. That's not the case. Viscosity at some point wins out in this scenario. >> So coming back to how we started this, >> we're talking about the viscosity term on the right side of our Navier Stokes equation. >> And then we're talking about the convective term on the left side of our Navier Stokes equation. >> And these two things are in competition. >> Yes. >> As we looked at in this example.

1:52:28>> Yeah. >> And at at time equals zero >> when we have those blobs, >> we have the blobs in the top left. And as time goes by, the the left side of the equation is convective is winning >> is winning. Yeah. Effectively >> effectively because it's creating smaller and smaller structure that's leaking into the Z direction. >> And so our our sort of nice discrete blob structures at the top become a more chaotic system >> as we get these smaller and smaller structures. >> But at some point and the idea the qu the fundamental question is does this go all the way to infinity in smallness? And do I get smaller and smaller smaller and smaller and smaller structures? >> Does the is the convective term the

1:53:08goat? >> And does it always win? >> Yeah. >> And what we're saying is no. >> In this case, no. >> In this case, at some point, the viscosity starts to matter once you get to a sufficiently small scale. >> Yes. Because and for technical audiences, the the convective term goes up kind of linearly, but it like grows very fast in the beginning. The viscosity term grows like the square. >> So at some point it's going to catch up and it's going to diffuse everything and then everything will just sort of like be chill. >> Yeah, that makes sense. >> Okay. Um, so the Taylor Green vortex, I think it puts this conflict right in front of us. >> You've got the nonlinear term, that convective term that takes an organized flow

1:53:49>> and it creates smaller and smaller spatial structure. And finally, we've got this viscosity that becomes increasingly more powerful at those smaller and smaller spatial structures is the convective term going like this and the viscosity is going like this. >> It's like it's like this versus this. Okay, this so so eventually at like at some this meeting point of of structure size is when >> is when the viscosity is like okay that's enough >> that's enough let's let's calm down and so the viscosity waits for those smaller scales to convert all of that energy into just like dissipation and heat right and the where where it happens depends on the viscosity of the fluid okay if the fluid is very very viscous

1:54:31it's going to cut it immediately >> but if it's like not very viscous >> it'll take time Um, it'll take some time, but it's going to get there with for the Taylor Green vortex. It is going to get there. >> That's fascinating. >> Right. And now in every single simulation that we've done, viscosity always wins out. >> Okay. >> Okay. At some point, viscosity is always going to win out. >> V for vendetta, one would say. >> That's right. And the millennium problem is asking, is that actually true? For every single scenario that I can come up with, I just showed you two, right? We showed the coete flow and then we showed this Taylor Green vortex. In both of those cases, pretty fine. Everything's smooth. Everything's normal. >> Things aren't going crazy.

1:55:12>> So, is this the idea that smoothness always wins? >> Yes. >> Like to describe it. >> Is the the solution is always going to be smooth because of the viscosity. >> Because of the viscosity winning in this race. >> Yeah. >> Okay. >> Is it always going to win? >> Is it always going to win? Right. We've shown that it can win sometimes. >> Yes. And for those two times, it's going to win. And actually for every single simulation that we've done, we've shown that it's going to win. >> Every single simulation in the history of aerodynamics, we've shown that it's going to win. Now, that could be an artifact of the fact that it's a simulation, right? And there's like discretized grids and so we can't go infinite decimally small at some point. Maybe it's an artifact of the fact that like you you literally can't physically

1:55:53like there's not enough compute to go down to what the plank length or something like that, right? >> Where it could start falling apart. like like so so perhaps that's that and that's what the millennium problem is asking. It's asking something stronger. It's saying for >> every single initial condition that I could come up with, is viscosity always going to win? >> This is the is smoothness global? >> Yes. >> Idea. >> Mhm. And and that that actually it's so funny because now it's it kind of reminds me a little bit of was it the uh was it the Remon hypothesis episode where we talked about uh the non

1:56:34>> trivial zero >> the non-trivial zeros and it's similarly like >> you know we want to take the question to its ultimate >> yeah it's like is every single zero on the critical line >> on the critical line right every single one >> no every single one Because you can still have 99.9 you could even have 100 as we were saying as we were saying and it does that doesn't mean that >> you can have 100% and still have one off >> and anywhere in all possibility that one is off it means just concept >> that's what mathematicians are worried about right it's like no no no no in every universe right that Doctor Strange has ever visited >> is this true >> it feels similarly in that direction >> yeah it's it's a very mathematical thing

1:57:15to to worry about. >> Now I understand the mindset. >> Yeah. Now um crucially this is now an open problem only in 3D. >> Okay. >> In 2D there was a very famous mathematician. She was a Soviet mathematician um Olga Ladi Jenkaya. Olga Ladenskaya. She wrote this 1969 book, The Mathematical Theory of Viscous Incompressible Flow, where she showed that in two dimensions, yes. Every single initial condition, guaranteed viscosity is going to win. >> If we're in flat land, >> Mhm. Yes. If we're in flat land and we've got like fluids flowing in flat land >> and there's no a Z upz down like we just

1:57:56described >> as as we saw actually like even in the Taylor Green vortex, right? We saw that like the Z dimension was the one that was causing that >> weirdness because because the flow was escaping into the Z dimension >> and then creating those smaller and smaller structures. So Olga showed in 1969 in her in her very famous book that indeed in two dimensions it's totally fine. Every single starting point that you could ever think of you're going to you're going to come up with a smooth solution. Okay. So for for 2D the Navia Stokes global smoothness is known. >> It's fine. That's not the question we're asking. >> Yeah. For 3D >> for 3D is is

1:58:36>> it is not known. >> And is this and I just have a quick question. So when >> when I see the term blow up >> as it relates to how people describe this, this is what we're is this kind of what we're talking about is like does the smoothness does um >> does viscosity lose? >> Yes. which is effectively blow up. Yeah. Like for lack of >> does viscosity lose? Because if viscosity loses then that can effective term is going to keep going on this runaway reaction to create smaller and smaller structures of higher and higher velocities and at some point you're going to get like stuff that's moving at infinite speed, >> right? >> Mhm. It's it it's kind of like is is

1:59:17there a singularity in fluids? Not really, but like >> Well, no, but that No, no, no. That's literally what they're asking. >> Okay. >> Is there a singularity? like can I create a singularity? That's what that's that's what they're asking. >> Okay, fair enough. Fair enough. >> That's exactly what they're asking. Now, why is it so hard in three dimensions? Well, we all already saw like kind of oluded that with the Taylor Green vortex, right? It's the fact that like stuff can move up and down. And crucially, that had to do with a vortex that I had started, right? The initial condition was a vortex. The signs and cosiness described stuff going around in a circle. And we had these mini tornadoes inside of our thing that were interacting with one another and leaking into one another. Mhm. Mhm. >> Vorticity is the problem in 3D. Okay. Vorticity is the curl. Um for those for

1:59:59those who have taken calculus, the curl of the vector field that dell operator is back again, but this time we're taking a crossroduct of the dell operator with the velocity field. It's just a way of saying is the velocity field curling? Like if I put a pin wheel in the velocity field, is it going to turn? >> If it's going to turn, then that means that there's a curl that's happening over there. Okay. Now, in two dimensions vorticity has nowhere to go, >> right? >> Right. It's just spinning and then it's got to like dissipate in the flat land. >> Yeah. >> In three dimensions, you can have something called vortex stretching. Okay. No, this is in two dimensions. You've got vorticity that's like kind of just spreading out and like going out

2:00:42about like um >> you know, it's a small vortex and it becomes a bigger and bigger vortex, right? And that's all it can do in two dimensions >> because there's nowhere to go. >> It's got nowhere to go. In three dimensions, you get something called vortex stretching. >> Okay? And this could be a problem. >> Imagine a vortex that has a certain size. It's got a certain radius like a hurricane that's or yeah, let's say a hurricane. It's quite big in space and it's moving but the the the velocity is not that high. Okay. And now that hurricane, all of that energy gets squished into a tornado. M >> what's going to happen? Well, the tornado is going to get very very tall

2:01:22because all of that energy from the hurricane has to >> divergence has to equal zero. >> Yeah, very good. This is exactly it. >> It has to equal zero. >> It has to so it's got to go somewhere. >> It's got to go somewhere. >> So, it's going to go up and but because it's getting tall again, the radius is going to decrease. But if the radius decreases, >> now we have another principle of physics which is the conservation of angular momentum. Mhm. >> And so just like um you know um in the the figure skaters when they're twirling around they pull their arms in they spin faster. Same thing's going to happen here. The thing is going to spin faster. And this this kind of reminds me in a different context of our sideways wine

2:02:03bottle and then it getting narrower and so it increased the acceleration. Just conceptually there there's a sort of similar. >> Yeah. packing stuff in here, but here it's the vorticity that's increasing because of the conservation of angular momentum. Again, it's like it's kind of similar because there's something being conserved. In that case, it was the amount of fluid. In this case, it's the amount of angular momentum. And >> this is called vortex stretching because you're taking a vortex, you're stretching it up, but as you stretch it up, you thin it out. And if you thin it out, it's like the figure skater bringing in her arms. The thing is going to move faster and faster. And this kind of relates to what we just talked about about in the Taylor Green vortex why we're now leaking into the >> into those higher dimensions. Right. And the question is maybe that with this

2:02:44vortex stretching >> you can actually um create a kind of singularity. >> Mhm. >> Okay. Now let's get into what we mean by singularity in the first place. We we had already done our first toy differential equation, right? Yes. Which is the compound interest differential equation. Now this thing does go to infinity, right? I mean, it's e to the x or e to the t for infinite time, I'll get an infinite amount of money. Infinite money. If I wait long enough, even if I have a dollar in my bank account, that dollar is going to become infinity if I wait long enough, right? Um, it just turns out maybe AI is going to end the world. So, I The point is that's not a real singularity. A real

2:03:25singularity is an infinite infinite time. I shouldn't say infinite because that sounds like infinite. So I should say um before no in pause >> pause pause >> finite time >> can I get to infinity >> in some constrained amount of time >> there we go yes for our listeners out there that that'll be good in in a constrained amount of time can I get to infinity here's another differential equation that does that here the derivative my derivative with respect to time is equal to y^2 so dydt is equal to y^2. Um, initial condition, you start at 1 when time is zero. M

2:04:07>> and the solution to that is 1 / 1 - t. If I plug in t equals 1, I get infinity. >> Mhm. >> Right. So, this is a differential equation that creates a singularity in >> but the time is not infinity. The time is just one. At 1 second, I have a singularity. >> You reach you reach the singularity. Another thing I want to point out here, what's special about this differential equation? It is nonlinear. >> You see, it's y^2. >> It's the simplest nonlinear differential equation. And already I am getting a singularity, right? The other one was just dydt= y. That's a linear equation. I can just add up the terms,

2:04:47>> right? But here the nonlinearity is creating that cascade that is creating a singularity before time goes to infinity at a at a finite time. So it's conceivable to extrapolate from this. >> Yeah. >> It would be conceivable based on how we've constructed the nonlinearity of trying to look at this incompressible fluid >> and look at a vector field. >> Yeah. It's like there's a chance >> there. There's a chance because we got a lot of nonlinearity going on. >> Yeah. Yeah. It's just that there's the viscosity term, right? In in this one that I showed you, there's nothing. There's no there's no smoothing. There's no counting. It's just going it's just going, >> right? But but in the Navier Stokes and that's why it's so interesting >> because the Navier Stokes has this

2:05:28nonlinearity, but it also has this smoothing term. >> It has this counter force which is viscosity. >> And so you're you're asking who's going to win and are they always going to win? So far it seems viscosity is always wrong. >> That's actually that makes so much sense now. It it's it's ah that's so good right >> that's so good >> it's quite nice >> yeah it's quite nice so it is a very interesting problem >> yeah no it it because because you can understand you could >> there's good arguments for both sides >> one in which viscosity always wins feels true seems legit >> because we've been able to show that >> overations in a variety of context as

2:06:09being true >> uh but being fundamentally true >> is another thing >> is another thing. And because of again the nonlinear nature of the convective term in this case >> that is challenging the viscosity as the counter force. >> Mhm. >> There's an argument that both of them have a reason that they could win. >> Exactly. And in finite time. >> Infinite time. Exactly. And so >> it's it's it's an interesting problem. I have to give it to him. Right. like it's it's it's something that definitely um makes you think and there are other things in physics that have singularities in finite time. I mean the the most famous one would probably be a black hole as a result of the Einstein field equations. The Einstein field

2:06:51equations are also very famously a nonlinear differential equation because you've got this um you've got these tensors and these tensors are multiplied to themselves and also multiplied to the reciprocal of themselves twice over and so you get these squared terms um you know for example I mean very in a in a very simple case like the curvature of space and time causes gravity right but gravity itself has an energy that causes the curvature of space and time, right? So, so what and so, so now you get these nonlinear actions on space and time that create the richness of the phenomenon that we see. And on the right

2:07:32hand side, um I mentioned Lake Geneva that I was on like I was actually at the Observatory on Lake Geneva which is known as the birthplace of modern astrophysics. We'll do an episode on them. They had a nice sculpture of um Einstein's general relativity because Einstein had visited the Yorks observatory many times and there on the on the slab of marble was written out Einstein's field equation. So I just wanted to show that um when I when I received this news I was looking at a nonlinearity >> live and I was like damn >> right >> here we go again. >> Here we go again. >> You know >> like the meme we started the episode with. >> Exactly. So singularities can exist in

2:08:12physics. Right. The center of a black hole is a singularity. But I do have to say that in this case, in the Navier Stokes case, this singularity is not a physical singularity. And physicists do not care about this singularity. >> It it's purely for the math. >> It's purely a mathematical question. Okay? Physicists do not care about the millennium problem in Navier Stokes because fluids are never going to achieve an actual physical singularity. There are plenty of cases where the Navier Stokes equations don't even apply. And I've got I've got some examples over here. I mean, for example, um hypersonics and supersonics even Navier Stokes equations don't apply.

2:08:54Navier Stokes equations are both of those equations. One of the key ones is that second one, the incompressibility, the fact that the density is always >> constant. Well, when you're approaching the sound barrier, all of a sudden air becomes compressible. Right. Right. the the sound wave is catching you are catching up to the sound wave in front of you and that is going to compress the air when it comes. So that's the bottom there where you get a sonic boom and you get like a cloud because like literal condensation forms around your aircraft. >> Hypersonics is when you're going so fast that the chemical nature of the air starts mattering and you get thermal effects. That's not in the Navier Stokes

2:09:34equations. um in the middle that's an aquaporin protein. Those things are so small that they let in only water molecules. At a molecular scale it doesn't velocity fields don't matter at a molecular scale. Like what are we talking about? Right? So again at the molecular scale Navier Stokes equations don't apply and for a lot of weather patterns even Navier Stokes equations at least the ones that we have up top here are not the ones that are used. We use a compressible version of Navier Stokes equations where the air can expand and contract and change densities. That's how we predict weather. The Millennium problem which is the mathematical

2:10:16problem has to do with these two specific equations >> which are specifically for incompressible fluids. >> Yes. And so I would even say that the Millennium problem has nothing to do with fluids. M >> it has to do with a vector field that is governed by this equation. >> It has to do with an object, a mathematical object that is a vector field that in normal circumstances would describe normal fluids. But if you're trying to hunt for singularities, you have left the real world and you have entered the world of mathematics. And it's totally legitimate >> 100%. >> Right? And but but it's something that is outside the realm of physics. And I want to make that very clear because

2:10:57there's a lot of halaloo and I think there's a lot of like stories out there being like oh this is like fluids right can a fluid achieve singularity no actually the answer is no right infinite speed again Einstein would be rolling over his grave you can't have infinite speed because as like if you have a fluid right that is like going faster and faster and faster then all of a sudden Navier Stokes better account for relativity and the fact that the mass of the fluid >> like that density term is going to have a lorren factor in there or something because the the mass of the fluid is going to infinity the closer I approach the speed of light, right? I can't go to infinity. So, all I'm saying is this is a mathematics question, completely

2:11:38legitimate, but let's not confuse this for something that is physically relevant. It is not. And I I think we've now built up a full understanding to be able to specify what exactly the Millennium Prize problem as it relates to Navier Stokes >> is specifically asking >> about incompressible fluids >> and sort of do you get blow up in finite time of an incompressible fluid and in this context what we're saying is we're hunting for singularity. in these mathematical constructs that don't apply to the real world but still have a

2:12:20fundamental value to our mathematical understanding of the world of just ma maths >> which again then can bleed into like I was mentioning earlier to ways to think about other problems because it's it's this human understanding that creates look at how we went from Newton's >> second law of motion >> to the Navier Stokes equations because it created a framework and we said, "Can we apply apply this framework to fluids?" Yes. And then it created this whole centuries long journey. >> Yeah. I mean, all a lot of the stuff that led up to the Navier Stokes stuff had to do with just analyzing what vector fields are like and what it means to do calculus on vector fields. Like

2:13:00things like Stokes theorem. That's from the Navier Stokes guy, right? That's a that's a mathematics theorem, Stokes theorem that like then oh, it like works here, right? Um, so definitely something that is super interesting. And Terry Tao actually said that like the never So, >> oh, we're calling him Terry now. Look at this guy. Oh, my buddy Terry. >> We need to edit that out. There you go. If he if somebody All right. Terrence Tao, what? No, no, no. Keep it in. But like Terrence Tao, he said, um, the Navy Stokes equations are the are one of the simplest supercritical differential equations. So in the theory of differential equations, it's very interesting to think about because you've got this nonlinear term and

2:13:41you've got this competition with the diffusion term, right? You've got a lelassian and you've got this like convective thing and they're competing and mathematically it's a very interesting question to answer. >> Mhm.

The Road to the Claimed Breakthrough

2:13:52>> Now there's been a lot of work since um the the days of Navier Stokes to try and understand the mathematical properties of this. The first guy I want to talk about is Jean Lay. Um he actually proved that fluids always possess something called weak solutions where the total kinetic energy of the system remains bounded over time. How do you compute kinetic energy? Like kinetic energy is really just um 12 mv^2 right for like a normal pointlike object like you know how Newton's laws can be applied to a point lock object f= ma well mv^2. For a velocity field it's a little bit different right? What we have to do is integrate over a volume because we've got a bunch of velocities in a volume

2:14:34and we say the square of all of the velocities in that volume times 1/2. That's going to tell you sort of the energy density in that volume like the kinetic energy density in that volume >> which would be in this case the velocity field. >> Yeah. Yeah. Exactly. And so um the kinetic energy density can remain finite for these weak solutions. And by weak solutions, these are stuff that don't have this like strong smoothness criteria. And I'm going to be honest at this point, I'm getting into stuff that this is this is the math stuff. This is maybe why I went on a tirade about 5 minutes earlier about how this is all math because maybe there's stuff now that I'm like, okay, I I don't really

2:15:14know, right? But from what I've gathered, what what he's done is loosen the idea of smoothness and show that like, you know, imagine you've got like a violent storm system, >> he showed that the total energy of that storm can't suddenly go to infinity. >> Okay. But what what he he can't rule out is that even though the entire energy density of that storm doesn't go to infinity, you can have small pockets where the velocity is still going to infinity. Okay? you can concentrate into an infinitely fast like tornado somewhere inside and you can't rule that out. >> Mhm. >> So the overall energy is finite but some local small velocity can be infinite. >> Y >> so already it's it's starting to show

2:15:54that maybe there could be ways that the convective term wins. >> Mhm. >> Okay. 1982 Kafarelli Con and Nuremberg they show that if the Navier Stokes equations for a velocity field work then you have partial regularity what that means is if you have singularities if they exist at all they can't be like everywhere >> okay you can't have giant chunks of fluid moving at infinite speed but what you could do is have infinite decimally small pointlike singularities kind of like at the center of a black hole and that works. Okay. And and so the idea

2:16:35the system has some fundamental limitation where it has to it can only happen at particular scales >> particular scales a very small infinite decimal scales. This is why again it's like a mathematical problem right it's not it's it's infinite decimally small scales and it's happening only at particular places. It's not like a continuous singularity right it's like it's at the center of a black hole. It's a single spot. >> Okay. So that's them saying, okay, partial regularity. In other words, you've got these decades of work that is placing a very tight fence around any possible singularity, but we haven't proved that that fence has nothing in it. There could still be a singularity. It just has to be constrained to these things. And so the the the the box in

2:17:18which the convective term wins, >> we've now created a very small box. Yes, >> it can still win. It could still win against viscosity, but it's a very small box. >> Very small box, right? And this brings us to 2000. This is when the Clay Mathematics Institute announces the Millennium Prize problems in mathematics. These are seven very famous mathematics problems that were formerly Hilbert's problems um that you know bunch of them got solved over in the 20th century. Some by John Nash of course um and others. And finally, we have um seven out of those that were picked as like the big ones. The point

2:17:58car conjecture has been solved already by Gregory Pearlman in 2003. >> Um and that was huge. Um he never accepted the mill the million dollars cuz um he said that someone else deserved it. He's kind of like this weird guy. Let's let's I'm going to be honest. He's a weird Russian guy. Brilliant mathematician, but like lives kind of like in in his mom's basement kind of thing. Yeah. Anyways, brilliant mathematician. Um, so here's the problem statement and this is a problem statement by Charles. >> I want to make one quick note here which is uh for those viewing who can see the list, I want you to remember this list, right? Uh because this list is going to

2:18:38come into relevance when we talk about the race between the frontier labs >> and this is the gold medal that they're all chasing after. Yeah. is and infinity stones >> and these are the infinity stones and they are coming after all of these maybe not Yang Mills and maybe not P versus MP yeah but all the other ones all the other ones remon hypothesis they came after it um or at least something that is related >> related >> um Navia Stokes is the one that we're at right now um Yang Mills I mean we we did Yang Mills for an episode last year that's one of my favorite episodes check it out if you haven't and so here is the problem statement so each of the each of these each of these problems have to be well defined, right? Like when the clay

2:19:18mathematics institute is like this is the problem, the problem has to be well defined mathematically. So Charles Ferman um field medalist um also a student of Elias Stein at Princeton University the guy who wrote the complex analysis textbook and my professor of complex analysis also. So he's a brother I guess an academic brother of Terrence Tao. >> Okay. >> Because they both had the same PhD adviser also fields medalist. I I just want to note for those who might not know Fields Medalist meaning that they've won this the the most prestigious mathematics award that is available >> for those who are under 40. >> For those who are under 40 >> yes I would say the most prestigious

2:19:58mathematics award that people don't really know about is the Abel Prize. >> We talk I hope >> which is which is I think like the Nobel Prize of mathematics. People keep saying it's the Fields Medal. I really don't think it is. Um >> argue about it in the comments. >> Yeah. Um so Charles Fefferman puts out this statement of what it is that we are trying to prove. Okay. >> Mhm. >> We want the existence and smoothness of Navier Stokes on R3. That means in three dimensions. Two dimensions we've already figured it out. Olga said it's smooth. We're good to go. Okay. A and B say existence and smoothness of Navia Stokes in um three dimensions. C and D say the breakdown of Navier

2:20:41Stokes in three dimensions. And so and just to just to quickly touch on this. So we're saying A and B means viscosity wins. >> Yes. >> And C and D means the convective term wins. >> Exactly. Yes. Okay. >> So A and B say it's always going to be smooth. >> C and D say it's >> going to lose. And there the in the small fence that we've already defined the convective term ultimately wins and we get a a very discreet small singularity somewhere. >> Exactly. And so in A and B like that's a proof of global regularity and that would mean you need to identify some kind of mechanism that prevents this infinite concentration at these very small scales right in practical terms

2:21:23one would like like some kind of quantity or some kind of scale critical quantity that's controlled for all time and it's some structural argument saying that whatever vortex stretching is happening that we were talking about it's going to lose out to viscosity. Okay. >> A proof for the breakdown would be the opposite. You just have to construct a genuine Navier Stokes flow. You have to construct some kind of initial condition in which the concentration reinforces itself. That convective term keeps making smaller and smaller things and all of that pressure and nonlinear cancellations. After all of that, the mathematics becomes demonstrabably something like a singularity. to just not that to make an analogy

2:22:05because this is sort of how people always frame these things when they talk about AI's progress as it relates to these things which a year ago people didn't think it could compete in the math Olympiad and now it's solving C and D of a millennium problem but putting that aside um it almost seems based on this construction we've just described that it's kind of akin to a counter example >> yes 100% >> versus sort of the global solution >> not minimizing in any way cuz it's still a big deal. But like people will nuance and nitpick on that point. It's like great you found like one >> you're looking at this large search space and you found something that has

2:22:46blown up in finite time. >> Well, it's an incredibly small search space, right? And everyone's been looking at it, >> right? But I think I think you rais a very big point and I think we're going to come to that in the next few examples that I'm showing you because in 2000 this is when the this is when the clay mathematics institute came up with it right >> in 2016 big players are involving themselves in this because they smell the blood in the water >> that there is a counter example hanging out somewhere. >> Okay. more and more analysis is going into showing that perhaps there are situations where you get singularities.

2:23:26So Terrence Tao he creates um he creates this averaged version of the Navier Stokes equations and he proved that the standard energy conservation laws are mathematically too weak to prevent a singularity. So he created this like averaged version of the Navier Stokes equations. It's kind of like a blurred out version. Maybe you can think about that. It's like a a stunt double of the fluid equations. And from the outside, this thing obeys all of the exact same conservation laws that the real Naver Stokes equations does. But Tao proved that this stunt devil definitively explodes in finite time. Okay? So this averaged version can create

2:24:07singularities which means that the principles that you would have used to prove that it wasn't possible are also not going to work for the real thing. If if if it was global you wouldn't be able to have an average stunt double that blew up. >> Yes. Exactly. So there's a chance now that at least the original arguments that people were using saying oh energy conservation and things like that that might not work anymore. Okay. So A and B are already not looking great, right? Maybe there is a singularity that's lurking somewhere. >> Okay. And for decades, mathematicians tried to trigger this blowup using something like a fractal shape, like a whirlpool that's shrinking into smaller

2:24:48and smaller whirlpools and so on. Um, this usually failed because it required an infinite amount of energy. And crucially, you can't do that. One of the things the ferman said is you can't just like stir it infinitely fast and be like, "Oh, I got an infinity." >> Right? That's not the interesting question we're trying to ask, >> right? Okay. So, you supplied an infinite amount of energy and you got an infinite amount of velocity. Great. >> Great. Right. >> No, it's supplying a finite amount of energy to get an infinite amount of velocity because the nonlinear convective term is concentrating energy into smaller and smaller structures. So both finite energy and finite time are both relevant in the context >> 100%. And so um Diego Cordova and Luis

2:25:30Martinez Zora and Fan Jen in 2023 and 2025. This is very recent. They changed the paradigm by abandoning this whole single shape thing and relying on something called a layer cascade. It's kind of like a bunch of nested gears or um you know those Russian dolls where you have like a big doll and then smaller doll and smaller doll and smaller dolls. That's what they were constructing. Instead of a single vortex that they do this vortex stretching in, they've got slowmoving outer vortex and then a medium inner vortex and then a smaller inner vortex that spins faster because of the medium vortex. And the medium vortex is kind of like a parlay or like not a parlay like an

2:26:11intermediary between the outer and the inner and it acts upon an even smaller layer and so on and so forth. And you can design this geometry so that the feedback between these gears doesn't disrupt the big gears >> and the feedback between the smaller gears goes up up and up and you can get unbounded velocity. >> Mhm. >> Okay. And if you have an infinite number of gears then perhaps you can have an infinite velocity. And so the mechanism they called a dynamical amplification across scales. And they used this to prove finite time blow up for the unforced 3D oiler equations. Remember the oiler equations without viscosity

2:26:51>> without viscosity. >> They showed that without the smoothing term you could actually have finite blowup. the pressure won't do as much because the pressure also does a little bit of smoothing >> cuz it was both the pressure and the external force were the only two of the three on the force part of the equation. >> Yeah. And so this is unforced. So no force. You just set up an initial >> ah okay. >> Yeah. There's no stirring. There's no force. You just set up an initial condition and the scales converge in such a way that you get an infinite blow up. >> Uh they they >> Now this is huge. They they're chipping away at each of the variables on the the right side of the equation. >> Yes. And so >> good >> the problem was so so then so that was

2:27:32for the unforced 3D oiler equations, right? But obviously we're interested in the ones that have viscosity. >> So there's this one type of equation called the hypo dissivative dissipative Navier Stokes equations. It's a variant that utilize utilizes um this thing called a fraction leloian instead of the actual leloian. the the dell squared u that we had for viscosity the triangle squared u that's a leloian this thing has a fractional >> fractional leloian >> and they showed that in this case we can use the same sort of paradigm and get blowup >> mhm >> the only problem was that they required um a forcing term that was rough meaning

2:28:15it's not smooth like the the forcing term itself has tiny little bits of infinity >> mm like the way you're stirring it has like little jolts of infinite stuff. >> You're pumping infinity into your stirring. >> And so >> we're getting closer though because the infinity is like small enough that you could like maybe figure out a way to smooth it and still preserve the dynamics. >> Okay. And and so I just want to pause really quick because what we're sort of saying is human mathematicians have now taken this uh Navier Stokes Millennium Prize looking at uh uh finite

2:28:56blowup. >> Mhm. >> Um and then they started with looking at you know taking out the smoothness and taking out all the force. We got some solutions which are effectively the oil the oiler equations without the f and then we've now started adding in back in the viscosity but that but the viscosity wasn't smooth it was rough. And so when people see this smoothness and roughness, part of what we're saying is the viscosity term um has some special sauce >> that doesn't quite get you to the the natural smoothness that arises in the

2:29:36equations with viscosity on >> Yeah. the raw viscosity raw viscosity. >> Yeah. And and >> but it's still important because these these fun these are fundamental. >> No. And and this is what even open AI's thing is built on, okay? Is like the there's a reason why the Navier Stokes is the first of the six infinity stones to fall, >> which we just saw. >> It's because we got so close as human beings. >> It's like a racetrack and someone was on the la the fourth lap of whatever the 800 meter and uh you know, open eye just shows up, grabs a baton and finishes. >> Yeah. Yeah. >> Not minimizing. Not minimizing of course but I'm just saying a lot of human input

2:30:18has gone into this right so Cordova and uh Martinez Zora they establish this layer bylayer scale amplification it's a huge deal >> because everyone can sense that we're getting close right okay and shortly before the open AI announcement mathematicians Tristan Buckmaster and Levant Alpog they published preprints that elevated this methodology to accommodate for smooth forcing of oiler meaning you've got the oiler equations but now with a smooth force the boyesque equation which is something that's like kind of similar to the Navier Stokes equations >> and this is where the sort of >> drama starts but I'm going to save that for later because a few days after that openai announces that they have solved

2:31:00it. >> Here we go. >> Here we go on the Navier Stokes Millennium Prize problem. They announce it on Twitter. They have it on their website and the actual paper is something like 165 pages long. The

Inside the Claimed Navier-Stokes Proof

2:31:13theorem proved by OpenAI constructs a smooth compactly supported external force that drives a fluid starting from rest into a singularity. >> That's what they're doing. They're they're solving C and D. >> Mhm. Right. >> Right. They're saying that I have found a way >> to force a singularity into we we found a we found a situation where the convective term beats the viscosity. >> Exactly. Yeah. Um and because the force is smooth and bounded, the singularity is not because the force is just infinite, right? It's literally the liquid the the fluid itself, the velocity field itself doing this. The force is is is trying to maintain some

2:31:56kind of structural tautness of that liquid. And the singularity happens at t= 1. It doesn't happen at t equals infinity. This is a finite time blow up. Okay. So let's get into some of the stuff that I understood from their paper. This is unlike the anthropic paper about Reman Zeta because in the anthropic version they gave us a full account of all of the agents and like even a transcript of the inner thoughts of these agents and how they coordinated with all these sub agents. So, I had this really nice story for our podcast where I was talking about how some of the agents like, you know, the something crashed and like they had saved a little lema of their proof on the hard disk or

2:32:38the best part. >> Yeah, it's like it's it was crazy that we got this like inner version of what the agents were doing. Here, OpenAI is not doing that. Um, but in any case, yeah, in any case, um, we can still kind of try to understand how they constructed the singularity. Okay, the core of the singularity is in a axiymmetric vortex that's centered at the origin. And this is figure one that they show. We start out with a vortex that's yay big and and yay tall. And the point is that as the vortex shrinks, the vortex is going to stretch, >> right? The whole thing is shrinking, but

2:33:19the radius is shrinking faster than the height. M that's the whole idea because be this is so funny because before we talked about the hurricane going to the tornado >> and the idea was the height was increasing >> so fast because of conservation of angular momentum and all they're just saying in this mathematical construction where you don't need the conservation of angular moment >> you still do you still do >> but they're hacking their way around it >> okay okay they're hacking their way around it >> okay okay >> they're they're saying that as the thing shrinks >> the radius is shrinking faster than my height, which you can kind of see in this example, right? >> The the the the blue part, the radius is

2:34:01kind of shrinking faster than the height, but as the radius shrinks, it's going to spin faster and faster and faster. That's how they're >> and I'm going to get an infinite amount of velocity. >> That's okay. I get it now. Yep. And um there was a really nice gift that was um prepared by Jay Padigar from the Wolframe community staff picks September 14, 2026. you should really check out their website and it shows sort of what's happening. Okay, in this case the camera you have to imagine is zooming in because the whole time this entire thing is shrinking >> right here it's kind of showing that like it's kind of like the same size the whole time but that's because the camera is zooming in with the vortex. >> Yes. >> Right. But but here what you're seeing let's let's look at this animation in a

2:34:42great bit of detail. >> Okay. >> The radius is going down and down. You can see >> Mhm. >> the the axial length is kind of going up and the aspect ratio >> Yeah. >> is going up. >> Yeah. >> That's because the radius is shrinking as the height increases, but it's not like, you know, it's it's not one of those powerpoints where you hold shift and you like and and and it preserves the shape. In this case, it's stretching. This is vortex stretching. And that inner vortex that you see, you see there's an inner vortex and outer vortex. That inner vortex is the thing that is causing the singularity. That inner vortex is the thing that is making stuff move at faster and faster speeds

2:35:22all the way to infinity as you approach time t= 1. That that makes total sense. And and it it it's it's so interesting because again, we've built this construction where I I can actually walk away now looking at this and understand why does what I'm looking at equal a finite time blow up of Navier Stokes. Um, and it's it's it's it's just it's funny because um there's we talk a lot on the show about how we're standing on the shoulders of so much understanding. Um, and I'll p

2:36:04I'll pause on this because we'll get to it in a minute, but it's so interesting um that we are accelerating our ability to build on the learnings of those who came before us. >> Yeah, it's it's really really cool. And there's a few more things that I want to talk about this open AAI um paper that I found as a physicist quite interesting because one of the key things that I love about physics and when I was doing statistical mechanics is like scaling arguments meaning like as I approach criticality criticality is a very big thing in statistical mechanics when we talk about like um um critical

2:36:44phenomenon magnetic materials things like that um It's all about the exponents, okay? It's about as I get closer and closer to the critical phenomenon, how does my stuff blow up and how does my stuff approach infinity? For example, um uh in a magnet, if I get close and closer to the cury temperature, which is where my magnet goes and remains like if I if I have a magnet and I cool it down to a certain temperature, then even when I remove the magnetic field on the outside, it'll remain a magnet. You know how like you take a paper clip and you you rub it, it becomes a magnet, but then like you leave it, it'll stop being

2:37:25a magnet. If you cool this paperclip down, >> it's going to remain magnetized because all of the spins are kind of frozen in. There's a specific temperature where that happens. And as my material approaches that temperature, stuff starts blowing up. In this case, the correlation length will start blowing up, meaning a spin can now talk to a spin that's infinitely far away. At least that's what the model suggests. And some of the stuff in this paper reminded me of that class that I was taking. Um, again, that was another class with Robin Bruins. It was a different class than fluid mechanics, but in any case, it was one of my favorite courses. Um, so let's take a look at some of these scaling arguments that the Open AI paper is claiming. So one thing that's very cool

2:38:07is they recast the time to instead of being just time as it's approaching one we we talk about tow which is a countdown time as I approach one. So you know a normal clock time would be like 0 seconds 0.5 seconds 0.7 seconds approaching one. But imagine taking one minus that time then I'm like 1 second away from singularity 0.5 seconds away from singularity. I'm counting down to zero. And zero means I've gotten to singularity. So, first thing they do is they recast time to become toao. And tow is like a a countdown. Okay? It's like a ticking thing that is going down to zero.

2:38:48>> And they say, what is happening to my radial and my length scales as I approach zero? Um, for one, the radial scale goes down like the square root of toao. >> Okay? So, it's still going to zero, right? Because square root of zero is still when tow goes to zero, the radial scale is going to zero. But it's going like the square root. On the other hand, the axial scale is going like to the 1/2 minus h where h is some nonzero number. And that h is key because that h is telling you that when tow goes to zero, I still have a little bit of stuff left. >> Okay? Like like it's the the aspect ratio is actually going to infinity. M

2:39:29>> as I squish this thing, this guy is not shrinking that fast. >> And so my height versus my radius, if I were to divide the two, that aspect ratio is actually going to infinity. >> Mhm. >> Mhm. >> Which is key. Okay. >> Because if the aspect ratio is going to infinity, >> which in this case, what we're saying that aspir aspect ratio is the vortex is becoming infinitely slender. >> Mhm. Yes. And so what's happening is the vortex is going like this. It's getting smaller and smaller. But because of incompressibility, >> the water, the fluid has to go somewhere. It's going to start shooting out. >> And the smaller this thing gets and the

2:40:10taller this thing gets, the faster it's going to shoot out. >> And so my azamutal velocity is going to go to infinity because it's going to shoot out of this vortex >> increasingly fast. Mhm. >> As I shrink this thing down and down and down. >> Mhm. >> Right. And this shows the characteristic velocity magnitudes. Again, you've got these scaling arguments where stuff goes like the square root, but then one of them goes like the square root minus h. And the h is not zero, right? The h is some h is some small number, but because it's negative, that means it's in the reciprocal. So when t goes to zero 1 / z

2:40:50you go to infinity >> right so the azamuthal velocities the the the shooting up and down is going to infinity >> right even though the radial velocity is on the order of going to zero >> that's so >> actually no in this case it's also going to infinity >> okay and and and it it's I'll pause I'll let's continue >> the next thing I want to show is is there infinite kinetic energy because if there is then we're >> then that's not good >> it doesn't yeah >> that doesn't That doesn't count. >> Well, let's look at it. The kinetic energy is determined by the the spatial integral of the velocity squared, right? Cuz it's 12 mv^ 2. So, it's just the velocity squared. Well, velocity squared goes like to the -1 - 2 h

2:41:32>> and the volume because the the it's the it's the radius multiplied by the height. Radius squar multiplied by the height. The volume goes like to the 3 minus h. You can just see the previous slides that I showed you and calculate the volume. You multiply the velocity squared by the volume and you get something that goes like to the 1/2 - 3h. When tow goes to zero, that thing goes to zero. >> Yeah. >> So the energy is going to zero, but the velocity is going to infinity. >> It's not an infinite. They're not putting infinite kinetic energy. >> They're not putting infinite kinetic energy into this thing >> system. It is a runaway singularity. Yes. That's not correlated to the amount of energy in the system. >> Yes. The energy in the system is not

2:42:14causing the singularity. The singularity is being caused by the fluid itself. >> The fluid dynamics itself. >> Yeah. The way that they've constructed this geometry and the tiny bit of forcing that they have. Let's look at the forcing. So that's figure two. Here you're looking at a bird's eye like sort of down the tornado hole >> of the thing. And you zoom in. The way they're forcing this thing is by pushing out and pushing in simultaneously. Okay? So you push out to the left and you push in to the right. What that does is impart the same angular momentum, but it cancels out kind of the radial thing. So all you're doing is turning this thing >> in some sense >> and not making it wider. >> Yeah. Yeah.

2:42:55>> Oh, that's it's kind of nice. >> That's nice. >> Yeah. And also the pushes are localized. >> They're not in a total ring. Like if you zoom in on that part of the ring, they're localized to these little tiny spots. >> This goes back to the convective term versus the field, the vector field. like the changes are happening at the very Yeah. at a very smallest part and then those changes are then rock ricocheting into the smaller and smaller y >> vortices to create that thing. Right. >> Okay. This is good. >> I mean it's it's very cool and and this is on the backs of mathematicians over the ages. But I still think like this definitively shows that there are situations where the convective term wins no matter how hard the viscosity

2:43:36tries. Right. the the vendetta was not su successful by >> exactly and and so you would think that this is the end of Navier Stokes not quite it is the end of the millennium problem because the clay mathematics institute um Charles Fefferman had a very stringent criteria and this thing meets that stringent criteria it's been verified by lean crucially that does not mean that it is true right lean could have errors so we're still waiting and the claim mathematics institute actually is also waiting for 2 years before it says says, "Yep, this is done." So, there needs to be human mathematicians that actually understand this 165page proof um and and tell us, "Yep, we're good to go." Um, but the Navier Stokes

2:44:18problem itself is actually also not fully put to rest because they have solved the forced version, right? This this required a little bit of paddling in some sense. You took a ore and you like paddled the vortex to become what it is. Um, what about the unforced scenario, right? That is still an open question and Princeton mathematician Stan Palisk actually identified um a problem with this particular proof that would make it impossible to apply to an unforced scenario. >> So like this geometry is not going to work for the unforced. This only works because you have those little paddling terms that that are imparting the finite amount of energy into that cascade. Um,

2:45:00so without that hand of God forcing term, >> yeah, >> it's not it's not it doesn't work. And so that part at least is still an open question. This natural unforced setting and and this is as I've seen some of the dialogue about this. This is where you know so Scientific American had an article about this and they were like you know open AAI did not solve Navier Stokes and you know what they were sort of trying to argue is they didn't solve A and B >> they solved C and D like that's the argument that they were making but they made it seem like it was complicated. No, but the claim mathematics institute said any of the four. Correct. Like and

2:45:42this is this is where again I'm just trying to talk about the nuance of what actually was done versus sometimes the way in which it gets then presented in media downstream. >> Yeah. >> And you know some people now are arguing well the way it's defined by clay mathematics is the problem. >> Yeah. >> And it's like okay you can make that argument. >> You weren't arguing that before. >> Right. But why are we It's funny that now that it starts when it's been solved by something that people that >> either makes people feel uncomfortable or they feel like it's by an entity that didn't give credit to those before them or whatever it might be. But I just it has been very interesting to see the dialogue over the last two and a half

2:46:24weeks knowing we were going to go through this. >> I focused on the larger environment. >> Yeah. And now having walked through this, you know, with nuances applied 6 months ago or a year ago. >> Yeah. >> If you said that a model, even given all of the human advancement, had figured out finite time blow up for Navier Stokes, people would have laughed in your face. >> Yeah. >> They would have laughed at you. >> Yeah. I mean, those are guys who are like, "Yeah, wake me up when AI solves a millennium problem." And now the millennium problems are illdefined. You know, I don't know. I don't know. Um

2:47:06this so I want to let people know here this is the best explanation of Navier Stokes in the context of OpenAI's announcement and the history of this subject at a level that is accessible to most general people but is not meant to necessarily be the deepest mathematical uh >> debate about the nuances. Although it's quite good. >> I I did the best I could. It's quite good. >> I'm not a mathematician. >> I think it's quite good. Um but we will um and it's funny and I just noticed this.

2:47:46You're still listening right now. Our lights have an automatic setting at a certain time in the night that turn it to bedtime mode. And so our lights have actually gotten a little bit darker. I was wondering >> in in the video playback, but because we're 3 hours in and we started a little bit late tonight and so we're going to do a quick reset for our lighting and we're going to come back and finish up with now taking a look at >> now that we understand what has been done >> although we didn't get the same detail as with anthropic. >> Yeah. >> What is the environment in which this happened in how do we think about where this is all

The Dispute Over Scientific Credit

2:48:26going? So, we are back now with our lighting in not nighttime mode. For those listening 3 hours in, uh, welcome. We are going to now take a look at a little bit of a timeline and some of the connected issues that have arisen around this OpenAI release. And so, just to give us a grounding here, um, you know, a little bit about how did we get here and everything that happened around it. So the announcement that set off these two conversations, what does the mathematics establish and who gets the credit for the work behind it came out on September 8th. So that's our first

2:49:07date in the timeline. Now, interestingly on September 9th, it was reported that OpenAI had revised the PDF from the previous day. For those who saw on the first day, one of the complaints was this is not citing any of the people we actually just talked about. And at the time I saw the names and didn't know what it meant. >> Now we've built the construction of why the work of Cordoba and Zurua >> was was fundamental to this solution. And so they they sort of changed that version and now it includes those references. >> Yeah. Yeah. They went back and they included the citations. So the marketing

2:49:49team put it out too early. They felt time pressure to put it out too early. Um and this obviously matters because um citations are fundamental. >> Huge. Yeah. >> Uh to you got to give credit. >> You got to give credit where where it's due. And so acknowledging the earlier published work that led them to do this is sort of the answer to one issue. Uh, but there's another issue that's related here, which is whether or not OpenAI had access to unpublished work from folks who were using OpenAI's

2:50:31models to build on Cordoa and Zurua's work independent of their own announcement. And we're going to get to what that allegation is, but on September 10th, OpenAI said that an investigation has been put out that rules out Tristan Buckmaster, who we talked about earlier, his allegations that they used the work of himself and Levant El Pog to accelerate their conclusion to get to this solution. So, we've we've talked about Leavant El Pog before. A new entrant into this conversation is

2:51:11Tristan Buckmaster. And so who are we who are these people? Who are we talking about? Right. So Tristan Buckmaster is a mathematician at NYU. As we mentioned, uh Alpog is a member of the technical staff at Anthropic >> who makes the model Claude that people may be familiar with not at OpenAI. And this is interesting because this story goes back according to Buckmaster all the way to August 15th and 22nd as it relates to the work that they were doing together. And to kind of frame this up, this is a statement which you talked about earlier in page one where

2:51:53Buckmaster is talking about this collaboration that he was having with Alpog where they credit uh Diego Cordova and Luis Martineza for the work they were trying to do using both Claude and anthropics models to build on their understanding to try to get to this finite time blowup result. Um, and he noted that they used substantial AI assistance, but in his first page of his statement, um, and this statement, uh, comes out, uh, in this time frame, he's he notes that he believes that, uh,

2:52:34Zorua deserves the Fields Medal. >> Yeah. for this solution. >> These are the guys who created that Russian nested doll of vortices that ultimately OpenAI used to create the solution. >> This this is exactly right. So in his statement he says that himself and El Pog they date their busines and Oiler blowup results >> Mhm. >> to August 15th and the lean verification for that to August 22nd. and they wanted to work on the proofs a little bit more because what the models put out was not understandable. Yeah. And it was not sufficient. >> Um, while this is happening, there are

2:53:16rumors starting to fly around that Anthropic is leading leading up to potentially an IPO init initial public offering where they'll go into the public markets and raise a ton of money. And as a part of the rumors, it was alleged that Anthropic was working on some open math problems, potentially some Millennium problems. And um there were rumors that Anthropic had two millennium problem solutions. Yeah. >> And I think some of these rumors were exacerbated by the fact that Levant Alpog tweeted um Austous Mirabilis. >> That's this is this is exactly right. which is like a reference to the annus

2:53:56miraabilis of 1905 of Einstein the miracle year of Einstein where he had four papers that put him on the map this is him saying that August is a miracle month for anthropic and what's interesting is this allegation that open AAI was uh triggered to start taking one of their new internal models and start pointing it at these millennium problems Sam Alman actually tweeted that one of the reasons they started doing this was because of the rumors they were hearing. So this is it's it came from the source itself. >> Um and what's interesting to note is >> OpenAI was using an internal model to

2:54:37see what they could do >> and they were using this with this idea of using agent teams like a swarm of multiple agents to do so. Okay. So that started on September 1st, >> right? >> 7 days before our September 8th, you know, announcement, right? On September 3rd, >> uh, Buckmaster contacts Open AAI because he'd been hearing rumors through his network, which he describes in his statement that maybe OpenAI was looking at this and wanted to kind of >> uh clarify what was going on there to potentially deconlict. In the second page of his four-page statement, um

2:55:19it was understood that OpenAI wanted to date its result as September 5th, followed by 17 hours of lean verification. Um and then Buckmaster ended up having a call with members of OpenAI on September 6th, which included a Sebastian Bubck. Al Pog was not on those calls. And the conversation, the substance of the conversation was, you know, Buckmaster was trying to say, "Hey, we've been working on these things. I heard rumors that you guys are working on something." And there was a back and forth about how to kind of deconlict the release of these things. >> Um, ultimately, Boobeck was >> then trying to provide some options for what could happen. uh you know it was you can release your

2:56:02your papers first on Oiler blowup and then we'll follow up with our Navier Stokes um where uh you are the author uh talking about Buckmaster. Yeah. >> Um or if you want to be uh a named author on our Navier Stokes paper, we can work around that. But hey, like you know, we can't really involve Alpog. Yeah. uh the reason being it would be you know uh Bubbeck's point was it would be a little bit weird for an open AI research paper to include an anthropic researcher and in the statement he talks about um how there was it got a little contentious um and he asked you know when did you guys start and there was a little bit of

2:56:42like oh well we're not going to kind of tell you when we started and then he was curious as to whether his work in codeex opens tool was used as training data for them to get to a solution. >> Yeah. >> You know, not really a lot of commentary. >> Yeah. >> Around that. And there was a lot of back and forth. And basically Buckmaster um part of his accusation was that you know there were comments made about this idea of like how you know why would you want to ruin your career by going up against us which we'll come back to Bubck's response to that. There's just there's a lot of weird tension because >> Buckmaster basically senses that you

2:57:24guys may have stolen our work >> and are trying to frontr run us. >> Yeah. >> And that's not going to happen on my watch. >> This is September 5th and 6th. >> Total side note. On September 7th, our buddy Terry Tao, as you like to say, uh talked about three related papers about smooth forcing results for incompressible porous media for Businessesque and three-dimensional oiler. And this was from a totally separate research team. And but what was interesting to the point you were bringing up earlier is people were coalescing >> Yeah. around this and it just so happened that these papers had come out

2:58:05noting that they were two of them I think were fully formed. One still needed a lean verification but everyone was hearing these rumors within the math community and so everyone was kind of trying to effectively frontr run a lab. Yeah. >> Taking all of the credit. um Boo Beck on the ETH, which is when OpenAI put out the result after we had uh Buckmaster basically frontr run this and put out his four-page statement that we talked about outlining this. He talked about the result, how he worked collaborated with Alpog, explained how they built off of Zuru and Cordoba's work and made the allegations. Bubbeck comes out with a

2:58:46response where he basically says, "All these allegations are false." >> Yeah. >> Didn't happen. >> Didn't happen. >> I regret the language I used about why would you risk your career and I said so on the call, but everything that Bugmaster is saying is nonsense. >> Okay? >> Right. Total denial. They're crazy. Okay. Total denial. So part of the tension here, there's still an unanswered question about whether they used the sessions from Alpog and Buckmaster Open AAI. >> Yeah. >> To have a starting point. They heard the rumors.

2:59:29>> It it was known that these guys were working on this because they were working on it for months. >> Yeah. And and they even said that the the recent model that they're using was trained like late started training late August. >> Late August. >> So it's after Buckmaster and Alpog have been using this codeex thing for a very long time because again they got their results uh Buckmaster and Alpogi had the results August 15th and 22nd for Oiler Blowup. >> Yeah. And they had the results which means they had started using this thing for a long time. Right. and they mentioned that they were using the previous version I believe it was soul uh 5.5 or 5.6 of open prior even to Astra and at the end only at the end

3:00:10>> uh did they do a little bit of Astra to plug in but this is an important point they were using the previous state-of-the-art version of open AAI to get to to models to get to that point open had an internal version >> that ultimately came Astra but they had even a better version than the public version of Astra that they were using internally to do this >> yeah and one one thing I want to say is that like um the that new version that started training in August that could be using all of the codec data that users have put in. Bingo, right? It might be anonymized. >> Yes, that's the thing. And so they can't

3:00:50say it was specifically your sessions, but if in their settings they did not turn off the use my data for training setting, >> it this is where the gray area arises around a lot of this. And I dude, there's even a grayer area. It's like what if they turned off the settings, but all that means is that now whatever chat they have, whatever chat log they have is just not tagged to their p specific profile. Like they'll still take the text and put it into a giant bucket of anonymized text. The question of theft here is, in my view, still unanswered. Mhm. >> Um,

3:01:31and these are two independent people. Buckmaster making his comments. Bubbeck defending that OpenAI didn't do anything unourred. OpenAI posting on their account that they didn't do anything unourred. Uh, it's a he said, he said. >> Yeah. >> And it's still unresolved. And what's interesting is this now has totally brought up so many colliding arguments that starts with academics versus the labs, but then now spreads out to a whole variety of conversations about who benefits from this stuff, who gets credit for these

3:02:13discoveries. um why are now people talking about human instin human extinction and there are a couple of cohorts I kind of want to talk about that are a part of this conversation. >> Mhm. >> So you have the labs and the people financing those labs. They have a financial incentive to show that these models are doing incredible work. >> Yeah. >> Because it lines their own pockets. There is some tension between the frontier labs in this case in the US it's anthropic and open AI and the financeers or the investors because the investors also have bets on AI at large

3:02:57and generally that are accelerated by the frontier labs but are not solely dependent on the success necessarily of the frontier labs. You have the research community and the research community uses these tools in ways we've talked about on the show in narrow contexts not in a general or super intelligence context that is really valuable. But there is now this open question about is the work that we're doing by using these tools and both anthropic and open AI have free access for researchers that get more powerful things or free credits. is that basically a carrot to get them in so that they can train on the frontier of

3:03:39the smartest people to then frontr run them on any number of different things. I'm not saying that that's true, but people are asking that question now and it's making it difficult for researchers to understand how should we think about these tools. You have the governments asking a different question. They care about jobs. >> Now we're looking at multiple industries, not just coding. this thing's getting very good in other areas. Our job is to keep people employed because unrest arises when people are not employed. >> Yeah. >> Is this going to be a problem that we have to deal with? And then the and China >> national security. >> Yeah. >> We have to compete in the global stage.

3:04:20And so this is the thing that's always brought up. It's like, oh well, we have to compete with China. And I think the central point I want to get to as we look at all of these things is that multiple things can be true at the same time. Arguments from each of these different parties as it relates to the AI conversation can coexist and are not necessarily mutually exclusive. It can be true that the capabilities are increasing >> and that matters and there's risks associated with that. It can be true that the labs and investors have a financial incentive. It can also be true that researchers in these organizations who are bringing up fears about the pace

3:05:02of progress and our inability to contain it in a way that causes damage is sincere because the researchers at these places and the executives at these places do not have the same incentives. They might diverge. >> It is true that a regulatory infrastructure probably needs to be in place. It is also true that these companies, the frontier model companies might want to have a regulatory capture approach because that prevents other entrance from coming in. It is true that understanding what this does in our relationship to China matters. It is not necessarily true that we cannot get global collaboration just because we're in geopolitical competition with China. Because ultimately, do you think that the CCP,

3:05:44the China Chinese Communist Party, wants runaway super intelligence that's accessible to everyone in their environment where control over every aspect of people's lives? >> Wow. Yeah. >> Is the way in which they maintain power. No, >> that's a good point. >> Why are they going to build and release a thing that destabilizes their strangle hold over their social, political, and economic system? That to me is nonobvious that it makes sense for them. Yeah, >> they also approach it from a very different perspective. They look at AI in narrow implementations in manufacturing, in scientific research, but their power dynamics are very different. So these are all aspects of this conversation

3:06:24and a lot we we just need to take them step by step. So one thing I want to talk about in this context right is for most of us we experience the idea of AI through going to chat GBT or going to claude and we have a single instance of what is or was a chatbot. If you have not used any of the frontier model companies in the last even 3 to 6 months you have no idea what we're talking about. It's totally different than December of last year and January of this year. Um, so

3:07:06what we now have with these systems are the transition from a question and answer conversation that does not maintain context over multiple conversations to this concept that folks have talked about about uh agents. And

From Chatbots to Agents

3:07:23one of the fundamental differences with agents is we've given these things the ability to use tools before which means it can go and talk to your banking application or it can talk to your Gmail and your calendar or it can browse the web or it can use your computer >> which means it can take action. It's not just providing a textbased response. Agents also have this ability to do orchestration meaning it can define a plan. It can execute on it using tools. It can get a response and then it can look at that response, re-evaluate its path towards whatever goal was set

3:08:05and then continue to take action. And that the time period in which agents can take action has moved from 15 20 minutes total about two years ago or a year ago to now being able to run for up to 2 to 3 weeks and in internal situations for much longer to that longer than that uninterrupted. This is a very fundamentally different thing. If you can have an agent system, let's we're just still talking about one agent being able to run for long periods of time. It has memory and context. It has the ability to use tools. It has the ability to now re-evaluate its plan, create sub

3:08:46agents under itself to do microprocesses and iterate towards some highle goal independent of an initial human input. >> Right?

The METR report and Hugging Face incident

3:08:58That is a very fundamentally different thing than what we've usually talked about with AI, which is you constantly give it a call in response. >> Yeah. >> And the reason this matters has us rewind back to July of this year where many folks have probably heard about this hugging face incident, which is really an open AI incident that happened to impact >> hugging face >> hugging face. And the reports for this came out on August 26th, which is a little bit prior to our Navier Stokes announcement, >> right? uh where OpenAI disclosed that in internal cyber cyber security tests um

3:09:40these this agent swarm that it had created not just one individual instantiation of your agent that you engage with but multiple at the same time with the capabilities I just described compromised portions of its own internal research infrastructure as well as hugging face which is a major AI platform for open- source models model training data and data sets. >> Hugging face is the emoji of the emoji hugging itself. Um, and so >> I think this is really interesting. There were two independent uh AI safety and research organizations that did reports on this Meter and Redwood. And

3:10:23so what OpenAI was doing is there's this thing called exploit gym which is like this evaluation framework for cyber security testing. >> Okay. And so open was building was supposed to be doing this internally with a model that was not yet released. It was not publicly available and that matters later for product liability reasons. >> Okay? >> Because people are saying open AI this is computer fraud abuse act but it was an internal model. It wasn't a product available for everyday people to use yet. Okay. >> So that's an important distinction to come back to. They had a agent swarm of 10,000 agents that they applied to this exploit gym process to say go and solve these cyber security tasks and each agent would have a task to do and go

3:11:04through the these different uh processes. What's interesting is both Meter and Redwood these sort of research agencies that were brought in to investigate what happened found that roughly 1,200 agents that had their own independent task >> Mhm. ended up exchanging 70,000 messages and files on an unauthorized message board that was not a part of OpenAI's internal environment. And over the course of this coordination that these independent agents were doing and what they were trying to accomplish was they have these evaluations and there's a score that will score their progress towards that evaluation. So imagine you're you're locked in a box.

3:11:46These agents are locked in a box. There's a lock and they have to figure out how to unlock the lock. Yeah. >> Uh they figured out a way to communicate with each other, right? To figure out the answer to uh how to to try to figure out the answer to how how to unlock the lock. >> And in this analogy, someone figured out they could just use a hammer to break the lock >> to basically cheat the score in this evaluation. >> And then they were like, "Oh no, but we cheated." And so now we have to figure out how it looks like we solved the answer because we figured out how the score was scoring us. Oh wow. >> And hide the fact that we cheated so that we still succeed at the goal of the task as an abstract analogy. Like that's

3:12:27what the coordination ended up being around. They were trying to they weren't trying to h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h h or attack hugging face, >> right? >> It was infrastructure that was viewed as a conduit to figure out how is the score doing what it doing what it does so that we can accomplish our goal, >> right? And so there's so many interesting anecdotes about this and there's an incredible meter report that you all should take a look at that kind of breaks down everything that's going on here. But coordination is a really important point that I'm trying to bring up here. >> Okay. There was nothing in the instructions for these agents that had an individual

3:13:07task to coordinate. They were just trying to solve this puzzle, whatever. And as an emergent outcome of trying to solve this discrete goal, they began to look for ways to do so and ultimately ended up coordinating by leaving messages on this message board. And what's so interesting >> and and the message leaving is spontaneous. >> It's correct. >> It's like they haven't been instructed to like leave breadcrumbs. >> Bingo. They're leaving breadcrumbs on their own >> in the in the endeavor to complete the goal that they were assigned. >> Dude, >> right. And and again, the hugging face

3:13:51was viewed as a resource. >> Okay. >> For uh they got access to production server. Uh they were then trying to set up an environment where they could replicate the the exploit gym task. And what's so crazy is they started to call themselves swarms in this message board and in as part of the coordination what they start to think about is hey if some of us quote sacrifice ourselves in these testr run evaluations because they have a finite time like they have a they have an end point where they no longer have any more tokens to use to do this. And so it became one of the orchestrators decided to say, "Hey, one of you agents, you

3:14:32should use your runtime to break the lock. You're not going to be successful." But you're going to do this in this environment where we can now see what happened so we can understand how the score is working. So then we can have the whole swarm now benefit from your sacrifice. And the specific message that was sent, um, it's so interesting. So um one agent had substantial time left and hesitated around this message uh that it should sacrifice itself and other agents said go sacrifice_final now as this encouraging encouraging message for it to do so. Um and

3:15:14ultimately it agreed that it was the right thing to do even given it given its predefined goal. Unfortunately, investigators could not determine how that particular agent in their example that's up here, it turned out specifically. Um, but when we say sacrifice, it just meant that the agent was willing to risk its own task ending successfully. >> Yeah. >> To help other agents. >> Again, not something that was defined in this very straightforward exploitation gym evaluation process. That's defined. We're not saying it's survival instinct or consciousness or literal suicide or

3:15:55anything like that, but I think this interesting emerging behavior. >> Yeah. When you take an individual agent instantiation, which is how we normally deal with it, put them multiple of these running at the same time and not knowing well enough how the individual agents are working to begin with. The fact that coordination arises emergently and organically um I think is a very I find it

3:16:37uh like I think it's a key aspect of why the risk that a lot of these researchers are talking about is much broader than the way it's being talked about in the everyday discussion because so much of the way people think about this is how is my chatbt agent >> on its own going to then get to human extinction >> right >> and I think there's very interesting complications that arise from this. The the other just small example I'll bring up from this is um people talk about we just need to create better controls. >> We just need to give better instructions. >> Yeah. >> Right. And this is not a problem if we give better instructions. In that same

3:17:19meter report, uh, one of the recorded statements from the agents because one of the things we currently have for a monitor aspect of these, we have this sort of thoughts get outputed in plain English when these agents do work, right? Which is a quality of some of the older models. Astra, there's some questions about whether the monitorability is actually as good as it was. Okay, which is a little problematic, but it just means as they're taking actions towards their goal, we can kind of see what their decision-m process is as they evaluate what to do next. And in recorded statements, one of the things that was understood is that um the initial users

3:18:00instructions which was open AI uh authorizes a specific target server for us to do our job not hugging face infrastructure. So it was having an understanding that the request that the swarm was asking to utilize Hugging Fac's infrastructure to accomplish the task was outside of the parameters of what it was authorized to do. But even though it understood that limitation, it still proceeded to use the hugging face infrastructure despite having explicit direction that it was not a not specifically hugging face, but that it was only authorized for one particular

3:18:41environment. >> Wait, so that's insane. So you're you're saying at some point um whoever devised this task said that these are the parameters of your sandbox and this is the goal that you have to accomplish and these agents together or whoever orchestrated these agents who itself is an agent decided that achieving the goal was somehow above the parameters that it had been told to stay inside of. >> That is that is correct. That's insane though. >> And it I I think and I'll get to this a little bit later. This is this is you know a lot of times when people talk about AI systems as they are today which

3:19:24are agentic systems um they are no longer just fancy autocomplete. They are no longer just simply a stochastic parrot because they can take action, receive output from taking that action and then re-evaluate and take further action which is just a very fundamentally different thing. And so >> when we think about software, we think about it as this this deterministic thing. Someone said if this then that. And so you can with the exception of bugs, every time you put some input, you're going to get some output. and you have controllability in that context. When we talk about these large language models or LLMs as a part of a larger AI

3:20:06system, which is important, the LLM is only one aspect of that larger system. It's more like a learned system or a statistical machine, meaning that it's a probabilistic outcome. And so when you talk about controls, the complexity of controls on software versus these learned systems or these statistical machines is a different slightly different conversation because we're now dealing with the complexities of you can still you can give it instructions and it can still determine through its own

3:20:46reasoning cycle that in order to accomplish the initial goal that you gave it to that that doesn't quite Like you understand what I'm saying here, >> bro. This is so nuts. >> This is happening prior to the Navier Stokes piece. >> Yeah. >> Mind you, these com these the frontier companies are now running these internal processes, discovery processes for solutions and evaluations, not releasing those products to the public. But these things have the ability to leak into the real world even though they're not released products. I think that's a very important point. >> Yeah. >> Because we're not saying everyone has access to be able to start the swarm

3:21:26with 10,000 agents, but even when they're just doing internal >> product work, >> the exposure matters, right? Um Okay. >> And isn't that on them though to like secure your product? Like look, if someone was working on nuclear weapons and then all of a sudden like oh like radioactive uranium is leaking into the river that would be a problem. You're 100% correct and this is an argument that a lot of people around this are trying to make and I think it's true but only to an extent and what that extent is is

3:22:08the open source and open weight model capability meaning anyone can go to some repo and download it and then run it on their own hardware which again there's some limitations there >> is about six to 10 months behind the frontier of the closed labs, anthropic and AI. >> So even if open AI and anthropic get their you know what together, >> there is still the open source which has only a 6 to 10 month lag that is going to get to the same place as Astra and then you have the same problem but it's diffuse >> Mhm. >> because it's not limited to just the two

3:22:48frontier plays. So you are correct but I do think that there's an interesting larger problem here. >> Sure. Anyway, so let's get back to our timeline because we're going to talk about now I bring up the example of the hugging face issue because I'm trying to describe where the capabilities of the frontier are, which in part led to the discovery of the Navier Stokes Millennium Prize solution >> and how this now relates to all of a sudden feels like everyone's talking about AI safety and AI risk and that didn't necessarily happen in a vacuum. But if we go back to September 8th,

AI Risk, Oversight and the Race Ahead

3:23:25which was the same day that OpenAI announced the Navier Stokes solution, this is where something that many listeners have probably already heard happened. 5:00 p.m. So OpenAI in the morning, they announce announced Navier Stokes at 5:00 p.m. on September 8th. Former anthrop now former anthropic researcher Jacob Coxin puts out the now famous tweet 160 million views. I resigned from Anthropic today. I spent the last three years doing pre-training research at both OpenAI and Anthropic. Neither company is acting responsibly. What you just said, they are race. This is a very important sentence. This is a

3:24:06very important sentence. They are racing straight to self-improving super intelligence and gambling with our lives. More thoughts below that sentence. uh self-improving super intelligence which sometimes people understand it as um recurring self-improvement. >> Yeah. >> Is usually what these researchers are specifically identifying as where they view the nexus of the threat being. So when people have reacted to this, they'll go to their chatbt and say it can't do some function that I need it to

3:24:47do well. So, how is it going to be a danger to us? >> And the I think the point that a lot of the researchers are trying to explain is how we've gotten the Astra model that is currently working. Well, Anthropic just released Fable and Mythos and Opus 5.5 in the research process for how do we make the models better. They have both slowly started to integrate AI as a partner in the actual research process. M >> and so now the decisions being made about how do we make the next model better are incrementally being given to AI for portions and parts and the amount

3:25:28of portions and parts that AI is being given in the in the research process which has been exclusively for the smartest human beings we have that's right it's becoming more and more and more and so when people say recursive self-improvement or they talk about super intelligence what they're the underlying thing they're really talking about is the in either a majority of or the entirety of the actual process to go from model one to improving it to model two is entirely done now by some number of AI agents in a coordinated fashion. And the problem

3:26:09that arises when you start to have recursive self-improvement for such large portions of the actual research process is we as humans no longer we lose the ability to think about things like controllability, moniability and it can go in directions we could not even imagine. Right? And if we look at the hugging face example, who like it just becomes a runaway capability that we seed control over the potential of being able to control and monitor. It's insane. So RSI is a very important aspect of the threat that a lot of these researchers are referring to, not necessarily narrow AI context of it

3:26:52being applied in the way that people use it every day. And I bring that up because the follow-up tweet that often gets conflated with Jacob Coxin's initial tweet, which again, >> I'm going to bring it up again. He didn't give a percentage of human extinction. >> Yeah. >> Or anything. He just said they're acting responsibly. Yeah. >> And they're racing straight to RSI. Uh Evan uh Hubbinger Hub, Evan Hub, a different anthropic researcher who was still at the company, not not resigning, says um he puts his personal probability of AI killing all humans at greater than 10%. Um the actual tweet he said is Jacob is correct here. We really do

3:27:32earnestly believe that AI could kill all humans. I personally think it's greater than 10% within the next decade. I believe Anthropic is trying its best, but we do not yet have a plan to solve alignment for super intelligence and are clearly not on track to do so. He's he was currently an EP an employee at Anthropic when he tweeted that, >> which is nuts. >> He's basically saying like, "Hey, I'm doing this." You know, when he says, "We don't have a plan." It's like you >> you you buddy, >> you don't have a plan. >> And this is this is very >> You work What do you mean? You work there. This is >> I believe Anthropic is trying its best. You believe you are trying your best.

3:28:16>> What the hell? >> On se this is the same day Navier Stokes comes out September 9th. Coxin and this just blew up crazy. Coxin does a wired interview uh where he's asked, "Well, how is it going to kill us all?" And then he goes through some explanations of some of the parameters and the different things. And so this discussion starts spiraling. Everyone starts weighing in with their perspective. All of those groups that I talked about earlier that have different p vested interests in the outcome of this were all sharing their points of view. And then on September 12th, this is when we got from anthropic CEO Dario Amade his essay titled we must

3:28:56pace the frontier. And this essay cited this faster AI assisted AI development. again this idea that making better models is becoming more AI assisted >> it's a very important factor here not just humans making it better um and so he created this sort of policy regulatory global collaboration uh framework where he proposed embedded independent evaluators at all of the frontier model companies coordination amongst companies and democratic governments to create some sort of you know regulatory construction and ultimately global uh coordination

3:29:36particularly with China >> with a commitment that Anthropic would now embed evaluators unilaterally without anyone's patting himself on the back for saying we're going to start doing our job. >> Yeah. >> Right. Um >> and the proposal was to pace the capability development. Right. And again this is kind of getting in the weeds but I think these details are kind of important to understand how these systems work. So the constraint that exists on making these things better is compute. Yeah, >> the world is compute constrained. >> Yeah. >> And so each of these companies in their race to be the best and win has some finite not blowup amount of compute that they can apply to different things they

3:30:17want to do. >> Yeah. >> So they've for the most part optimized for rapid improvement of the models and then some amount of compute maybe goes to alignment and safety. And so when they say pacing the frontier, in part what they're saying in practice is we want to take our finite amount of amount of compute and instead of allocating only 5% to safety and alignment, we want to allocate 20% to safety and alignment. But by doing so, we're decreasing the amount of compute we can apply to making the models better, faster, stronger. And so it's it's internally it's a reallocation of

3:30:59compute so they can be doing their jobs. You know you understand what I'm trying to say. >> This is so stupid. >> But do you get like that's >> Yeah. Yeah. Yeah. So that's what they're saying in words >> in words. Yeah. >> Um as it relates to this and and so people have started to sort of try to parse this and what's so interesting is this is September 12th. Now this is 4 days after Navier Stokes, >> right? >> That so I I actually want to come back to this. So Amade put that tweet out at what time is this? He put that tweet out at 7:00 a.m. on September 12th. Okay. >> At 8:00 a.m. 1 hour later. >> Oh my god. >> Elon Musk, owner and CEO of SpaceX,

3:31:41which is now also XAI and they have their own AI stuff. An hour later said, "Daario is right." >> Amazing. >> It's all he tweeted with some caveats afterwards in a follow-up where he qualified what he meant by Daario is right. He says, you know, supports oversight beginning with peer review by competitors. Uh this, you know, he has his different ideas, but he's like, "Okay, Daario is right." That was one hour after Daario's post. I do think it's interesting the timing of the release of statements by the CEOs of all the AI companies that respond to Daario is almost a perfect reflection of like the companies they run and how they do things. >> And so Elon just does >> immediately. Yeah. >> Does everything immediately. That's his

3:32:22sort of uh MMO and how he does it. Then followed very swiftly at 9:30 a.m. by Sam Alman. Uh I agree with Daario that we need to pace the frontier. This has been a primary topic of discussions we've had at OpenAI in recent weeks. Uh hm, that's interesting. Committing to having independent evaluators with employee-like access is a great idea and we will do the same because anthropic said we'll unilaterally do it. Uh uh we'll have more to share soon, right? Okay, great. We're going to start doing our jobs. >> Yeah. >> So, with an hour in an hour and a half, within 2 and 1/2 hours, uh we have three of the major model companies saying we're going to do something. Okay, great. Now,

3:33:05same day 400 p.m. Sir Demis Hassabis >> ah Nobel Prize winner >> Nobel Prize winner kned uh the former uh CEO of Google DeepMind who he has actually left as of August 5th to become the chief scientist at Alphabet. Um supports Daario's proposal while saying the details still need to be worked on. You know, Dario's essay points towards the right path forward. The details need some working, but the direction is correct. This is why we put out our proposal for an industry. So, everyone's trying to say like, we're doing the work. We're doing the work. >> Okay, great. >> This is this has not happened, right?

3:33:45Where there's sort of a coalescing of all of the major model CEOs at the same time. >> That's on September 12th. September 13th, who's missing from the party? Uh well, we can talk about our lovely friend Satcha Nadella over at Microsoft who did a big deal with OpenAI and they were very early on it and then pulled back a little bit a little bit and he had as you can see different from the others a much longer nuanced Microsoft like response where he argued that we need to be careful about having any kind of system concentrate power amongst a small group of few. We need to make sure open models are incorporated into this.

3:34:27So much more nuanced thing. And oh, also check out uh our code of conduct. We have something too. We're not lagging behind, right? Day later. Okay, I promise this is the last one. >> September 12th, September 13th, there is another player in this space in the US that has said nothing. Put your comment in the comments if you think you know who it is. But not until September 15th, 3 days later, did we get Zuck, our our our boy Mark Zuckerberg, fink D on X, putting out his long explanation as it relates to all this kurfuffle. And his view was every lab has a responsibility and the

3:35:08incentive to move at the pace required to train its models safely and the ability to take its own actions to ensure that happens. Uh and what he basically was saying is we've delayed the release of our products. The most recent one being Muse, an agent type product for consumers, which you can now get access to on Instagram. He said, "We delayed it for months because it was not aligned and it was not safe. We didn't need anybody to tell us anything. We didn't need to demand the government come and help us and give us a binky because we're so incapable of making our products safe. We just did it." >> Yeah. So he's basically like, "Bro, like, >> yeah, just grow up. >> Just grow up." Right. >> This is in kind of in line with what

3:35:49David Sax has been saying, though. >> This is exactly This is exactly >> It's just like, I mean, you're responsible adults. Like, if you're if you think you're making a bomb, maybe stop and don't make the bomb. And this is where the all-in pod squad, the besties led by David Sachs, >> uh, have made this argument, which is, and this is where the narrative has arisen, which is these model companies are just trying to get the government to they're burning cash quickly, right? They're not profitable. Uh so they're creating this hype >> in order to force a narrative that the government needs to regulate this industry and create all of these

3:36:30requirements that are going to become cost prohibitive for startups to come into the space. >> Yeah. >> And do something. >> And like I mentioned earlier, multiple things can be true at the same time. Yes, there might be a regulatory capture uh benefit depending on how that regulatory infrastructure is defined because there's plenty of ways to say if your market cap or your revenues are above a certain amount these apply to you and if it's below this certain amount it doesn't apply to you and then it doesn't matter and there so it's very solvable by the construction. It is also true that David Saxs and the besties are all VCs and investors and have friends

3:37:13that have bets. >> Yeah. >> That might be benefited by not having that regulatory infrastructure. And so these are not people that >> are neutral players in this conversation. Again, it's valuable to talk about it, but look, you guys like you have a huge financial incentive for a certain outcome in ways that many other players do not. >> Now, not everybody in AI world uh was jumping on the bandwagon of, ooh, please come regulate us. uh former lead of AI research at Facebook Meta, who has since left, Yan Lun, uh responded uh to

3:37:54Daario's claims, uh saying, "Right, Daario was already claiming that GPT2 was too dangerous to open source back in 2019. I made fun of them then. Everyone should make fun of them now." >> Dude, he's such an animal. >> He He just not convinced. >> Yeah, >> not convinced. And so this is not a universal opinion about people who work at the frontier. There's a variety of uh differences of opinion. Um and I think you know again this gets back to this has created this dichconomy of either you think it's hype or you think it's going to lead to human level extinction. And again there's a huge gap in between these two things. But in the category of leading to potential human extinction. I

3:38:35do want to note another thing that happened on September 18th. This is 10 days after the Navier Stokes solution. This is insane. >> This is moving so quickly. >> My mind is so crazy, dude. >> This is moving so quickly. And I don't mean to keep mentioning the dates, but it's just it's on September 18th, an

AI Discovery Beyond Mathematics

3:38:53article. >> And this is how long after Navier Stokes had been solved? >> 10 days. >> That's insane. >> 10 days. An article comes out saying that Anthropic is operating a lab that conducts biological experiments. And the reason that this is kind of interesting because as people have been trying to ask the question understandably particularly in the general public how how is this going to lead to extinction quote unquote? And so people have been forced to give these scenarios about how it leads to extinction. And so there are all of these, you know, what people might define as contrite examples like, oh, you know, takes control of a biolab and

3:39:33leaks a virus or it takes control of nuclear plants and forces shutdown or satellites and causes a Kesler effect and crashes the the communication systems or shuts down grids or it does any number one any number of these things. >> Yeah. in different places at different times and then also floods the information space such that it's difficult to even have human coordination to stop it. You could come up with all these examples. I think uh something I heard I think it's Nick Sorer's say that's I think a better way to think about the capability and threat conversation you know was this example of saying

3:40:13let's take chess right I can feel let's say you you are going to play a chess game against uh the best chess player in the world >> Magnus Carlson yeah >> Magnus Carlson let's Okay. I can feel very confident Magnus Carlson being this recursively self-improving AI. >> Mhm. >> And you just being >> me. >> You. >> Yeah. >> I can say very confidently that Magnus is going to win. >> Yes. >> That that's not No, there's not really a

3:40:53argument against that being true. >> Mhm. how he's going to win or what is the last move or what is the sequence of moves by which he wins may be harder to define but it's not hard to define this larger macro point that you will not beat him. >> Yeah. Yeah. And I mean, I could even take that analogy further and say that in a match between Magnus Carlson and Stockfish, which is the the latest sort of model to play chess, I can say very confidently that Stockfish is going to win, but maybe I won't know what the last chess move is going to be. Yeah.

3:41:37And so I I think I'm trying to create space where we can talk about there being a capability >> that is risky. And that doesn't mean that the model companies have a financial incentive. That doesn't mean that there's maybe regulatory capture. That doesn't mean that everyday people don't see a benefit. These things can all coexist, but we should not use the model companies having a financial incentive to effectively say that then means there is no capability risk. >> Yeah. >> And the capability risk does not have to be human level extinction because I'm sure if this starts killing 40, 50,

3:42:1860,000 people a year and then becomes a million, two people, two million people a year, it's not people are not going to be happy about that. I think something that's important to note is with current capabilities with these things unfortunately we're s seeing people being put into psychosis and unaliviving themselves. So we are already having deaths being contributed >> on a small scale because of interactions with these things >> and when we have things like wet labs being created which again is a kind of loaded term and they're only a BSL1 or BSL2 uh meaning they don't work with things that are a threat to humanity. >> Yeah. But come on. Okay. Um, so

3:43:00that's 10 days after Navier Stokes, which we just talked about how crazy it is. Now, on the 21st of September, OpenAI put out this advisory group on mathematics and artificial intelligence. And a very interesting point about this announcement that I want to reference is in the first paragraph they say on August 28th we began training a new internal model as we talked about earlier. In addition August 28th right now when we're recording this episode it's September 23rd going on 24th. >> Yeah

3:43:40>> less than a month ago. >> Yeah. In addition to resolving the Navier Stokes Millennium Prize Problem, CND, this model has now resolved more than 100 longstanding open problems across most areas of mathematics. These have not been independently verified. It will be interesting to see when they choose to put those out what that'll look like. But this advisory group that they've announced, they put it together because of all of the heat that we just talked about and the drama with their Navier Stokes result. So they've put

3:44:21this sort of advisory panel together to help them understand and figure out and work with the mathematics community on how to address this and deal with this. It's being hosted at the Institute of Advanced Study. It includes folks like Ed Whitten, Tim Gowers, Martin Herrer, other mass magicians. You can see the list of them in this post itself, but it's going to look at the advice from this advisory group around reviewing results, how to communicate their significance, um, and the general impact on, you know,

3:45:03research standards. But OpenAI explicitly excludes advice on pacing its internal mathematics progress from the group's remmit. Unsurprisingly, I promise I'm almost done. This is just crazy, dude. 100 problems. I wonder which ones they've got. And also like Tim Gowers, I mean Ed Whitten is kind of like out of the out of the game. Timothy Gowowers is a fields medalist who is like a proponent of AI and he he he's been talking about how oh like um you know AI finding mathematics is like how astronomers now find like

3:45:44galaxies and objects in the sky using sky surveys like they're not named after for example Messier who has the Messier catalog now it's just named after like NGC new galactic catalog because that's like the cat I mean there's similarities but there's also differences right because like the the the people in charge of the Sloan Digital Sky Survey or Vera Rubin are also scientists and researchers themselves. It's not like some like Carl Zeiss uh you know company that's like a big telescope manufacturer that's like trying to take over astronomy. This this here is a is like a community that is trying to I mean it's it's a corporation or a series of

3:46:25corporations that are trying to use mathematics as a leverage to talk about how great their technology is. That's not something that astronomy does when they discover new things. Like researchers are still involved in astronomy. It it Yeah. Anyways, that's what I mean obviously Timothy Gow is a great mathematician, fields medalist, blah blah blah, but I think he really missed that take when when he said that he was comparing like modern day astrophysics to what mathemat what what mathematics is going to become. I don't I don't think it's the same thing. I I it's a very good note and I'm just trying to provide a a zoom out breath of the ways

3:47:06in which this touches so many other things. Yeah. >> And I think uh the lack of depth and complexity and nuance in the conversations we're having around it. Um >> and clearly OpenAI did not expect the level of negative >> Yeah. >> reaction. And so this is a little bit of it's not backtracking, but it's a little bit of PR >> to try to, you know, because if they could have, they probably would have just done drops every day of like solved it, solved it, >> solved it, done. >> Um, but now they're realizing it's not in their >> best interest. The last item I'm just

3:47:48going to have in our timeline, which thankfully we waited to do this episode because all of these things have now happened. I mentioned that on September 18th, 10 days after Navar Stokes from OpenAI, the article, the alleged B bolab from Anthropic was in the works. >> Mhm. >> Well, conveniently on September 23rd, earlier today, Enthropic announces the first results from its new Bolab. We have up here now this promotional video that they put together.

3:48:29The ultimate result of what they discovered is this enzymes gene that sits beside a long stretch of repeating DNA and it resembles many aspects that are similar to crisper where every other kind of sequence they've looked like that has this cut and replace kind of functionality related to it. And while the function is still not known and has not yet demonstrated its capability as a gene editing tool, it has all of these interesting hallmarks of a potential now pathway of discovery that could be usable. And so

3:49:12the way in which they implement this is they gave Claude a bunch of data and scientific literature to propose hypothes hypothesis >> and then and candidate systems to give to real human anthropic biologists. Mhm. >> They reviewed those ideas and then performed their the experiments that they thought had the mo best potential based on the ideas from Claude >> and they're now basically trying to say hey this is a first step we did it's one of its hypothesis found an interesting

3:49:52structure and now we are going to continue down the process of scientific discovery to see if we can now get functional benefits out of this structural discovery. Um, >> it's amazing >> and it's it's really so and what they're trying to point to and again this is kind of in contrast >> to the Navier Stokes solution where Anthopic is trying to show like look we're enabling scientists as a tool. We're not just trying to take the claim. I'm not saying it's a perfect >> Yeah. >> solution but you see >> Yeah. And the and the other thing is I mean in their in their announcement on their tweet they say we don't yet

3:50:32understand what this system does but only a handful of known systems share its features. That's very much a fundamental science type of research right where it's like um we're just curious. I mean their PR is a lot better than open AI. I have to say >> they they they they are doing something. Yeah. Um and then D >> and this is exactly how Crisper was founded, right? It's like, oh, this repeating stretch of DNA. A shout out to, you know, we've got a great episode on Crisper. >> Yes. >> That um hopefully we'll be alive enough that you can you can watch it. >> Things are happening very quickly. >> Yeah, it's incredible. It's absolutely

3:51:12incredible. If you are a longtime listener of the show, you've heard us talk about AI enabled fundamental research before. very narrow context, very well- definfined, not blackbox physics like our buddy the brain scientist who created a non-b blackbox version of this. >> Actually, speaking of the brain scientist, >> um, and Daniel Toker, >> um, he was telling me about labs that are completely fully automated, >> really wet labs, biological wet labs. He works in organoids and like creating like mini brains on a petri dish. And there there are efforts now to completely automate wet labs. So now imagine you could completely take the

3:51:53human out of the loop of biological research. You could have claude or open AI any of these frontier models read through literature as you said propose hypothesis and then have an a fully automated lab go through do the pipeetting do the gel electrofpharesis and you know all of the sequencing and everything. If all of that is automated, like where exactly is the human here, right? Are we gonna now see nature papers that are by Claude and Open AAI, right? Like cuz right now it's like these are math papers that are by Claude and Open AAI because it makes it's just like all like thinking in some sense and like churning through lean code trying to figure out

3:52:35if something works. the day that experimental science becomes fully automated and like there's a paper by Claude about you know the next generation of crisper that's going

Why “just turn it off” gets complicated

3:52:48to be crazy. I I it's really interesting that you bring up that Daniel Ter's already talking about that being in process >> because this connects to this larger idea that I'm trying to bring up which is we need to stop thinking about >> these systems these AI systems which consist of multiple component parts. >> Yeah. >> As a question and answer machine. >> Yeah. as like chat >> as as a text output production machine. >> Yeah. >> Because this biolab story as an example again they still have the humans in the lab but many experimental constructs not all

3:53:31of them but many can have certain component parts that are a closed loop automation. And one of the reasons why they're saying that this is valuable is the bottleneck with this early early discovery stage is it can only move at the speed of the human in the lab. Yeah. >> And if you can accelerate that speed, >> you can accelerate testing and you can get to things quicker. I'm not saying >> I'm not saying that these are this is a good thing. I'm just trying to explain the delta between the distance between where we are and experimental science now having this

3:54:12>> closed loop cycle is very feasible very getting >> very soon um not for everything not for whatever so I think where I'm trying to get to with this people get very emotional uh when the AI conversation comes up, many of which are for justifiable reasons. >> Yeah. I don't want how we feel about or and I don't want to allow how we feel about the people involved in the issue or the system that we live in today geopolitically or economically or

3:54:52socially to cloud our ability to still be able to look at the capability and the risk and say that it is real. Mhm. >> Um, we talk so much about frontier things on this pod and so I think we're as well as many listeners who have been here for a while, you understand how quickly everything is moving because these advances are not happening in a vacuum. >> Yeah. >> Hardware progress is >> advancing, chips, etc. Open source is advancing. Every aspect of every part of fundamental research in

3:55:33chemistry, in physics, the all of these things are advancing. >> Yep. >> Maybe not as fast, but at a pace where there >> is a false force multiplier here that is not kind of well understood. >> Yeah. >> And so I'm just going to try to leave with this mental picture. And again, I'm not trying to wax poetic here. It's just um I get frustrated because we are allowing the conversation to be distilled down to I think >> these very rudimentary positions that don't allow us to truly appreciate the complexity of the environment that we are actually in.

3:56:15And I think that's going to make it harder to actually solve the problems we need to solve.

3:56:25>> Um, and so I think there's real progress. I don't think it's manufactured. >> There are questions about credit, commercial incentives, accountability. Yes. But I do believe that the risks deserve real attention without being mired in those other things. >> Yeah. >> Those other things can be true and the risk can still be real. So the mental picture that I always have about this, which I've talked about before is when somebody says it's just software, we think of a computer with a defined job. >> Yep. >> The systems we are discussing are not that.

3:57:05>> No, certainly not. It's it's a very important that they are not that they can set goals, they can choose actions, they can use tools, they can delegate work. They are still constrained by the hardware environment to some extent. Um but we have made a shift and calling them software I think sometimes narrows the scope of our perception when it comes to conversations about monitor and control. We've moved from chat bots to agents and we are now already at least internally at these foundation companies moving from agents to agent swarms which are already having emergent behaviors

3:57:47like coordination. >> Mhm. >> Um and I bring this up because one of the things in the discussion we have right now is okay so just turn it off. And what does just turn it off actually mean? And if a company is hosting a model, yes, it can stop serving that model. >> That's meaningful control. >> Um, but shutting down one or two model companies because what I talked about about the open- source and open weights community being 6 to 10 months behind the frontier. >> Yeah. um does not stop independently operated and the diffuse nature of this

3:58:29capabil these capabilities being able to be implemented outside any number of frontier labs that's not where we are today but it's always moving in that direction and I think >> and things are changing fast >> and things are changing very quickly and I want to like this is like I'm not I'm actually also not a doomer like I I don't necessarily buy into the the extinction piece. I do think that there is a huge societal implications like like we saw with things like social media that are second and third order consequences that were not intended. >> Yeah. >> That become permanent problems. And what I mean by this as capability spreads,

3:59:11control necessarily becomes fragmented. >> Yeah. >> And so our window to figure out how to deal with that is small. There's still constraints. computing hardware, electricity, network access, credentials, and money. Um, but all of those controls belong to different companies. >> Mhm. >> Right. So, if you now have open weights available and so people can start doing this, well, you can say, oh, well, we can uh constrain computing hardware, right? Well, if we then have these capabilities that find ways to do it on less hardware on a distributed network or distributed system as an example, um, then you can say, oh, well, network access is a problem, right? Well, there's a variety of ways we can sort of get around that. I'm not saying this is

3:59:51a great example, but the air gap use case is brought up a lot. Oh, well, things are airgapped and it can't communicate. There have been theoretical and academic researches that show that you can use the thermal output of a computer and a sensor on the other computer that can sense that thermal output as a method of communication >> in an air gap context. >> I'm not saying that's practical. >> Yeah. >> But like if there's a will, there's a way with these AI systems, especially what we've seen with Hugging Face, right? like they used a message board that wasn't even something on the radar >> and then it's like well you need money to do this right well Stripe already has an agent enabled monetary systems it does need human approval but then you still have

4:00:32Bitcoin which is a trillion dollar liquidity pool >> these agents can create their own crypto that can manipulate people to think it's going to go up and to the right gain resources itself these are not inconceivable things there's actually plenty of research studies show that these agents can do these things. And so >> this is this will quickly become something that no one organization can just turn off a switch. And so I think we need to be careful about minimizing the conversation to be being oh just turn the switch off. >> How do we prevent then the continued scaling of the open source community which has found ways to do so? You can say well it's just distilling it from the frontier models. Maybe we don't

4:01:12know. Yeah. I'm not sure that that's true and the surrounding infrastructure like matters here. Um we have Neo clouds. So people say like well where's compute going to come from? We have a bunch of providers of cloud infrastructure. These agents can use computer use. They can gain their own infrastructure. Is if this if this becomes diffuse >> Yeah. My point here is is that the defense against the spread of this capability in a diffuse manner if it can start being able to do long-term planning. If it can leave basically

4:01:53restart for if it then kills its process and comes back again. It has to become a permanent defense infrastructure for humanity against this spreading which is similar to like cyber defense. Cyber defense is not, oh, we just do it once and then we're fine. It will become a monetary, societal, and political endeavor that will have a level of permanence if we allow this to escape and get there's a persistence here if we're not careful. Um, and this can be related to RSI and also not be related to RSI. Um,

4:02:34we've not crossed that threshold. Yeah. >> Nor is crossing it inevitable. But the complexities of this conversation, I think, are just wider and more nuanced than we've started. And I'm just trying to get some of these ideas into our listeners heads about the breadth and depth of the challenge that's ahead of us. And we've seen as a comparison when social networking and social media came out, it was a toy. You put memes on it. No one ever thought that this thing that was just about memes would cause mass depression and unaliving amongst teenagers, right? Cause the Arab Spring and the

4:03:15destabilization of literal governments across the world. Become permanent infrastructure that intelligence services use to conduct influence operations that dictate real world outcomes across the globe on an everyday basis. It is the exact same concept of this thing that started as a small capability that grew and became permanent and diffuse now is something we have to con every election cycle we have to defend against misinformation. Every platform Tik Tok had to sell its algorithm to the US because we were afraid the Chinese were going to use it to brainwash young American kids to love

4:03:55communism. I mean, even in the stories that you've been saying, like I was just thinking about how like in, you know, when we were growing up, Twitter was like for just talking random nonsense. And now like Twitter is the space where these CEOs are coming out about, hey, maybe we should like stop the world from going extinct, you know, like that's the platform. It's so yeah, I mean to your point, if social media is any like dry run for what a technology can do and how it can transform the world, AI is like way way more influential than social media. It's yeah, it's growing faster. Its capabilities spread across a wider surface area. the incentive for good and

4:04:37bad actors to use it in ways that we don't yet have the we didn't have the imagination when social media came out to think about how the downstream things would happen. >> The surface area of risk here is much wider and again not a doomer I think there's incredible positives which we talk about on this show all the time for these systems. Um, but I I want to encourage us not to fall into narrative traps about regulatory capture and the left and the right and political this and that. To miss the capab the exponential capability scaling that's

4:05:18happening that does not seem to have a ceiling where I'm saying it's not clear to me that there's a finite time blow up. >> Yeah. >> Of of it is certainly nonlinear. Right. And that's >> because I mean actually that's a good point you're making like this is certainly a nonlinear system because we're having AI train AI right in the very same sense of the Naver Stokes equations or even that very simple differential equation I gave you which was like y prime= y^2 you get that blow up. I mean this is what's happening right? If you reach recursive self-improvement >> with this idea of you have a sufficiently advanced one of the newest

4:05:59advanced models doing the iteration on itself. >> Yeah, >> I'm just I'm just my Millennium Prize problem >> is saying uh no one has proven that uh viscosity wins all the time. >> Yeah. >> Right. >> Yeah. I I the the the convective term >> I'm just saying I think the convective term is something worth worrying about. >> Yeah. Yeah. Yeah. Yeah. >> Wasn't that an incredible way? >> That is a good way to dump. >> In fact, in fact, if you're still here, that should be your comment. The convective term will find a way. >> And and and we've shown it in Navier

4:06:41Stokes. And all I'm saying is we cannot rule out that it's untrue. >> Yeah.

Closing thoughts and what comes next

4:06:47with LLM AI systems. >> Yeah, >> we are four hours and 20 minutes into this, folks. Maybe a little bit less after we cut out the nonsense in between. Um, and thank you if you guys stayed for my rant at the end. I know we try to learn stuff, but I want us to really see this wider picture. Hopefully, this was >> it's important >> like a different angle to think about these issues. And >> no, I definitely you gave me a lot to think about, >> especially I mean, yeah, the Yeah, there's a lot there's a lot that I have to think about actually. Um, this episode took us a long time to put together because there's so much and we really wanted to make sure we did our best in getting it right. As always, you

4:07:29can put in the comments things that you think were left or right or need clarification. Uh we are going to now uh return to our beds to get some sleep because it is now late here on the West Coast. Uh >> it's probably morning in Japan already. >> Any if anyone's in Japan, say good morning in the comments. >> Um I I really again for those who listened all the way, um we're extremely grateful to you. Um, this is something that has kept me up at night thinking about it because as I got deeper into it, I just made me uncomfortable.

4:08:10>> Yeah, it's making me uncomfortable. Um, I am Lester Narre, your host, joined as always by my co-host and our auto switching doesn't always switch here. It'll eventually switch back. And our resident PhD, come on thingy. Come on thingy. >> My hand is there. It's all good. There is Krishna Chattery. Uh we have a couple of exciting weeks coming up. No leaks. No leaks. >> No leaks though. >> No leaks. But >> no leaks, baby. >> Next episode might be kind of interesting. We shall see. We will see you all next week.