Navier-Stokes AI Solution — Full Breakdown & Transcript

[00:00] So Brady, I don't normally have sugar in the tea. Obviously I'm sweet enough as it is, right? But I want to show you that vortex in there. So what we have in there as I'm stirring that tea is a vortex and we have a fluid flow. We know that the

[00:14] equations that describe that, they were invented by a French engineer, Navier, and an Irish professor by the name of Stokes. And there's Navier-Stokes equations. I'm just going to write them down to begin with. So there are, as follows.

[00:27] don't we just turn that, do you? Yeah. This is the Denavio-Stokes equations. Now, the reason I'm talking about this, because you may have heard of the Clay mathematical prizes. So these were seven mathematical problems

[00:41] that were set in the year 2000, and there's a million dollars prize for proving any of them. They're also called the Millennium Problems. Millennium Problems as well, yeah. They're developed by the Clay Masters Institute.

[00:53] And I think it was solved in the mid-2010s, with Poincare conjecture. there's another one which is associated with this equation which was solved seemingly very recently not by a person but by AI like any by open AI so

[01:09] apparently what happened was that they set around 10,000 agents working on the problem they sent 2.7 million messages there were 130 billion output tokens

[01:21] developed. The cost, I mean, I've seen estimates ranging from $6 million to $15 million about the cost of actually solving this problem. But what's most remarkable is this problem has existed, you know, for the best part of 100 years. And they did it in 88 hours. So

[01:38] yeah, make it that way you will. We can talk a bit more about that in a bit. So we can say what a few things in this equation are, what the ingredients here. So the most important thing is this U. So this U tells me if I take a fluid and I take it, so U is

[01:52] actually going to be a velocity field, so it's U of, it's a function of position and time, so position in the fluid and the time, and it tells you, you take a little, you know, you look at the fluid at that position, a little part of the fluid at

[02:06] that position, at that point in time, and ask what's its velocity, you know, what's its speed and what direction is it travelling, and that's what U tells you. okay P here is related to the pressure it's actually the pressure to density nu and this is an important thing nu here is the viscosity it's related to the

[02:22] viscosity of the fluid and F here is essentially the external force that's being applied okay all of its P unit density and stuff like that but essentially that's what those are telling you so was F the spoon in your

[02:34] so F would be the spoon yeah absolutely u is the speed that things are going around and appears the pressure in the system what some of the things that matter here for example obviously d by dt here is the rate of changes with

[02:47] respect to time of that velocity field this operator here this this funny symbol here this is like the gradient operation you can see it here as well and here this tells you how that thing is changing with respect to space okay

[03:01] so you know they're calculated to derivative operator but it's taking derivatives but but it just tells you how quickly things are changing with So this equation is sort of comparing how things are changing with respect to time and space, the flow of the fluid.

[03:14] So Tony, if I was the god of mathematics, and I had this equation, and I had your spinning cup of tea there, what could I do that the immortal couldn't do? What power is it giving me? Well, it allows you to predict what the fluid is going to do next.

[03:27] So, alright, for a cup of tea, it's pretty obvious, I stay on the tea, it's going to swizzle around a bit. Okay, not very exciting. But there might be more interesting scenarios where you want to do what the fluid is going to do next. is going to do next. For example, waves.

[03:39] You know, what's a wave going to do? If I shake a body of water, what's going to happen afterwards? Am I going to create a tidal wave? These sorts of things, weather systems, all fluid systems. And it allows you, given the evidence of conditions,

[03:52] how much forcing you put on it. It allows you to say what the fluid is going to happen next. And it's used all over, of course. Right way from sort of, you know, mundane settings of fluid flows on Earth to in astrophysics. So

[04:04] it's a huge range of applications. it basically tells you how a non-relativistic fluid is flowed. So it's not going too fast, but it describes the equations for that. And I should add one extra ingredient here that I should

[04:16] mention, which is important, is this equation here, this tells you something about how it's called the divergence of the flow. This basically encodes the fact that the fluid is incompressible, so you can't squash it, you can squeeze it. If you push it in one direction, it's got to go up in

[04:31] another direction. You can't sort of push it all into one space, it's an incompressible fluid. Tony, you mentioned that this equation is already used, like it has practical application. Then what's to be solved? What was the unknown?

[04:43] What was the prize? What was the grail? To some extent, it is a mathematical problem, and it's a mathematical curiosity, and I actually don't think that the millennium problem really is physically important.

[04:56] But let me say what the question is. Basically, take this equation, and you imagine a scenario where you start off with a fluid that's nice and smooth. Nothing, no rough edges, no big kicks or infinities or anything.

[05:10] Very nice smooth fluid. And you only apply smooth forces to it. So again, you're not giving it sort of infinite kicks or sharp kicks. Everything's smooth. So start with a smooth fluid and you apply smooth forces to it.

[05:23] Okay, then the question is, what happens next? And the sort of subject of the millennium problem is the possibility that eventually you run into an infinity, that the system blows up at some sort of singularity. Now does that happen or doesn't it happen?

[05:39] That is essentially the point of the millennium problem. Now you might look at this equation and say well how could that possibly happen? I start out with everything nice and smooth and I only apply smooth forces to it. How can I kick something into blowing up like

[05:53] that? Well there is something here. So the point is this equation is not a linear equation. do I mean by that? So you've got... you can see these terms here, so this has got a single U in it, this has got a single U in it, this one's got... this one's

[06:09] quadratic in U, so this is not linear, it's quadratic in U. Now it's precisely this term that can cause things to blow up. Non-linearities in equations can do funny things. So for example, in general relativity, in Einstein's theory,

[06:22] that is also a nonlinear set of equations, it's a nonlinear set of equations of the space-time. And what can happen there? Well, something can happen that's kind of weird, that's a bit similar to this scenario, right? So you can have a situation

[06:35] where you've got ordinary matter, say a star, so everything's nice and smooth, there's only smooth forces being applied, and that star can then collapse to form a black hole, and then inside that black hole, space-time breaks down and there's a singularity, okay? So that's

[06:47] an example, and that's the non-linearity in the equations that force that to happen. So these are the experiments of Hawking and Penrose. And in a way, it's analogous, right? It's possibility that you can have this term that can cause all this all this sort of drama okay so could

[07:01] that cup of tea suddenly collapse on itself or turn into some weird shape or yeah exactly exactly so so so but this is the important important ingredient here this term isn't just there so what

[07:14] there's a bit of pushing and flowing and through and throwing in this equation so you've got this term which wants to potentially try to blow things up and but you also got this term which is the the viscosity then And this kind of and what you notice by the way this has more of these gradient operators it got more derivatives So this viscosity

[07:31] term wants to try to smooth things out actually. It's actually, you expect it to become more, so as the system tries to blow up, you expect this guy to start to become more important at short distances and actually try to stop things blowing up. So there's a bit of a fight

[07:46] between these two terms and in fact we really, when we really think about this equation, there's a quantity called the Reynolds number, which we encode the sort of 2 and 1 between these two terms. So the Reynolds number, I'll write it down.

[07:58] So if you look at this term, so if we imagine the velocities that sort of have some typical scale, which I'll just call u, and then maybe the distances, the typical distance scale in the problem, I'll call that l,

[08:11] then that means the gradient operator, which is like a derivative, the rate of change with respect to distance, that's going to go like 1 over l. Okay, so if I look at this thing, then the u dot grad of u, just by the scaling, is going to, well, I've got two u's, so that's a u squared, and I've got one of these gradient operators, so that's over L.

[08:32] Okay, if I compare it to this guy, that's mu grad squared u, okay, that's got a single u, I've got two of these gradient operators, so that's a 1 over L squared, and I've got mu.

[08:44] Now, the Reynolds number compares these two, takes a ratio, essentially. So, if I take the ratio of these two, I just get u squared over l divided by nu u over l squared,

[08:56] which is basically l u over nu. Now, the point is, what this says is, if this number gets big, this guy is winning, and you're going to get a blow-up, typically.

[09:08] And if this guy is small, the viscosity is winning, and it will smooth things out. I mean, it's the pushing, pulling a frame between these two terms that generally controls what's happening. Does it matter what the solution is, by the way? Like, you're saying we don't know

[09:20] whether things can blow up or not, right? You know, is it possible? Does that matter? Like, if things can blow up, whereas in real life they can't, you know, that's an extreme, does that mean the equation's flawed in some way? So, the equation is definitely, the equation definitely is incomplete.

[09:33] So, again, this is where the analogy of gravity is quite nice. So, this equation, it's an effective description of the fluid. It's not altruistic. relativistic, so what that means is the speeds of the fluid flow are quite low compared to the speed of light.

[09:46] So there's obviously corrections you would expect to the equation if the speed got very big, because it has to become relativistic. The other thing is you would do expect that this doesn't apply on a molecular level, right? So there'd be some point where

[09:58] the particles and the molecules are interacting, that that description supersedes this description. So this is not a fundamental description, it's an effective description. Same with gravity, by the way. General relativity we expect to break down close to the singularity of a black hole. So

[10:12] this is going to get replaced by some quantum field theory or something. So actually even though you might have a situation where there's a blow-up, in reality that's probably not going to be

[10:24] realized in nature. It just means this equation ceases to hold. Okay that's the likely scenario. What did AI discover? What did AI find? What was the answer.

[10:36] Okay, so I should probably just describe what they because the Clay Prize has various scenarios involved and I should probably briefly describe that and take it into that. So when it comes to Navier-Stokes problem, the Clay Prize

[10:48] is split into four categories. These were laid out by a guy called Seth Huffman. Oh, I've interviewed him. Oh, yeah. Charles Huffman. Yeah, yeah, yeah. You've always met

[11:00] me. We're going to be talking about people that you've other people important people that you've met. People way more important of the engraving. Okay, so they're A, B, C, and D. A bunch of scenarios that could be associated with this equation, and they're not mutually exclusive, by the way. They're not negations of one another.

[11:14] They could all be true in principle. They're kind of separated into various parts. The first thing you have to worry about is the kind of fluid we're looking at, the boundary conditions. Okay, so two possibilities, in fact, A and C, correspond to an infinite fluid where,

[11:31] so it goes infinitely far away, but everything drops off nice and settles down very far away. A and C are both infinite fluids, so they just have smooth boundary conditions far away.

[11:43] B and D, that's where you put the fluid in a box and you apply periodic boundary conditions. So these are like periodic. So they're just separated by those, so just reconsider both possibilities.

[11:58] And then there's another grouping. The other grouping is where you put A and B together, and another grouping where you put C and D together. Now, what's the difference there? So A and B, again, with the two different types of boundary conditions, the idea is to prove the following.

[12:11] This is the sort of task that Clay has set. So you assume there's no force, and then the goal is to say, if you assume that the initial data, the initial surface,

[12:24] the fluid starts out nice and smooth, you apply no forcing, and you do not have infinite energy, energy stays finite, the solution, the fluid, will stay smooth for all of time.

[12:37] Okay? So it survives and it stays smooth for all of time. So it's smooth use. Okay. Smooth seems such a vague term to me, but it means no spikes or infinities or...

[12:49] Essentially, yes. I mean, there is a precise mathematical definition which is that none of its gradients are glowing up. But yeah, exactly. So not just the first reason, not just all of them should not blow up at any point.

[13:01] So basically this is saying nothing ever blows up. Okay? Right. So the idea of A and B is to prove those to be true for the two different types of boundary conditions. Okay? Now, it may not be true.

[13:13] Right? We don't know. The idea is to try and prove it. C and D is slightly different. A bit more interesting C and D, I think. So here you're allowed a force. So you're allowed to stay your T now. Okay? And now the idea of C and D is to prove that actually

[13:27] you do get a blow up. Okay? So you get, you blow up. Not just you blow up, but there is, you know, maybe the whole tissue blows up or something, right? So some quantities start blowing up.

[13:39] There's a singularity in the flow. Yeah. Well, what does it look like? What does this

[13:51] weird, wild blow up look like that they cooked up? Okay, so it's a vortex, right? So this vortex, it has like a sort of central region, a middle region here. And in the middle region, the fluid is flowing inwards.

[14:06] It's flowing towards the center. Okay? So you've got a fluid flow towards the center. Now, because it's incompressible, it can't all just pile up in the center. It's got to flow out again. Right? So it flows upwards and it flows downwards, out of the center.

[14:20] So you've got this radial flow inwards and a somatical flow as well. And then you've got it flowing up and down, out of it. Okay? So this is the sort of wild system that they have. Now, this can then be shown to blow up.

[14:33] So how does it blow up? So there are various quantities that are important here. So there's the radial scale, the radial length scale, so typical size of the system, how things change in that direction.

[14:45] And there the angular scale of the system so the sort of things that control how fast things change as you go around And of course there the radial components of the velocity

[14:58] and the angular components. Let's say that we're at time t, and let's say the system's going to blow up at some time tb. Okay? Right, how do all these things scale as you approach that point?

[15:10] The length scale of the radial direction, that's going to go like tb minus t to the half. okay and along the angular direction it goes like TB minus T to the half minus

[15:26] H where H is a small number it's between 0 and 0.01 so what does this tell you this tells you that this this vortex this system is actually shrinking more quickly along the radial direction than it is along its angular direction so it's

[15:41] squashing more along the radial direction changes are sort of squashing more quickly. That h there is going to be important, that little deviation away from how that guy's scaling is important. Now we can also ask about how the velocities change in this system. So the radial components of the velocity,

[15:57] this is going to blow up, okay, and this blows up like 1 over tb minus t, again, so the half, and this guy, the angular direct velocities, are going to go like

[16:12] the same thing but a half minus h, sorry a half plus h. Okay so what this tells you is is that the, so whilst it's shrinking more along the radial direction, angular velocity,

[16:25] the velocities along the angular direction are blowing up more quickly than along the radial guide. Now why does this matter? Well let's look at the Reynolds numbers. We set the Reynolds numbers which is the push and pull between the non-linear term and the viscosity

[16:37] term in the equation. That's what really counts and you want a non-linear term to win. That's That's how you get a blow-up. So let's first ask what the Reynolds number looks like along the radial direction. Okay, so we take LR, UR, divided by nu.

[16:52] Okay, well, you can see that's just going to scale like a constant. Okay, you get LR times the UR. They're going to cancel, and it scales like a constant. Okay, so the Reynolds number doesn't blow up, okay, along the radial direction.

[17:05] But let's look at the angular one. okay we're going to be interested in changes in the radial directions okay but we're interested in the in how the the u velocity the theta component of the velocity changes okay so this

[17:19] is going to go like l r u theta over nu and you can see when you plug in plug everything in that this is going to scale like 1 over 2d minus 2 to the h.

[17:36] And so what this tells you is that the Reynolds number along the angular direction blows up, which means there's a component of the nonlinear theorem which is winning. So it's not that the nonlinear guy always wins

[17:51] against all pieces of the viscosity. That's not true. It doesn't happen along the radial direction. but it does happen along the angular direction and that is why you get the blow up that's absolutely key how does that manifest itself on your little diagram there

[18:04] does the jet flying up and down suddenly reach infinity so what it's saying is so there's two things right so the shrinkage along the along the radius is faster

[18:16] so the shrinking quicker along the radius but also the slurriness is going faster, more faster than the changes in the rate of velocity as you shrink it.

[18:30] And interestingly, the energy drops to zero in the center as well. It's kind of weird. Even though you've got this wild singularity appearing. So that's what's happening in the core of the system. And that is absolutely the blow up that they have.

[18:43] What's unusual about the force being applied? You said, like, if you apply a special force, this will happen. What's special about the force? It's an absolutely brilliant question. And it is, in fact, the question that you have to ask. Because creating this singularity by itself, what I haven't said here is how I get to the boundary,

[18:58] how I get off to infinity or how I get off to the periodic boundary conditions. I haven't thrown that on, right? And I have to think about that. And that's the hard part. That's exactly to do with the force. Okay, so let's draw it now.

[19:10] So let's draw another picture. So I want you to imagine, so we've got sort of our wild core, I'll call it, which is this stuff. Okay, I've got this wild core of this madness going on in here, right? And then I've got very far away.

[19:23] so let me just draw it sort of very far away and I've got some boring exterior which is just a very, and this is what they have they have a very sort of boring flow very far away, so nothing spectacular here at this wild core in the centre

[19:35] but you can see there's this kind of annular region between the two and you've got that annular region in that annular region you have to apply the forces to get from wild core to boring exterior, now because

[19:47] the core is so wild, typically you would need to apply infinite or very sharp forces to this angular region and that spoils the rules of the game. So the real magic of the solution here is to do

[20:01] that, is to manage to sort of apply the forces in this angular region that are nice and smooth and still make the transition. The way they do it is really really clever they use the non-linearity of the equation itself. So what they do

[20:15] is they hit this annual region with tiny little pulses, okay, so they cause little fluctuations in the velocity field and the pressure field, tiny little

[20:27] fluctuations. Those fluctuations average to zero, okay, so they don't really change the amount of energy in the system, nothing spectacular there, but their their square doesn't average to zero, okay. Now because this is a nonlinear

[20:44] equation the square enters. So they average to zero, so the fluctuations average to zero, so you put them into the system, you're not putting in much, they're averaging to zero, but because of the nonlinearity of the

[20:57] equation, their square appears as well. Okay, so you can engineer it so even though you're only doing these tiny little fluctuations, average to nothing, their square blows up on average and it's that, they fake the fake force that

[21:11] way, that's how it's done, by these pulses. They're averaged to nothing, but their squares don't average to nothing, and that creates, that allows them to get from the wild core to the boring exterior. Can you give me an example of a bunch of forces that average to zero, but their squares don't

[21:26] average to zero? Yeah, yeah, it's pretty simple. So we use kind of a wave-like structure, right? So imagine a wave, it pushes and pulls in two different directions, so on average, you've got as much positive part, as much negative part, but you square them, two positives, and then averaging

[21:40] for zero. That's it. It's as simple as that. And then they just scale it down so that it actually gives the squared part an infinite contribution which drives the transition from the wild core to the boring exterior. That's how it's done. It's pretty impressive.

[21:55] With any big discovery we ought to acknowledge how we got there. I think you've seen these articles about climbing a mountain and you can celebrate the first person to climb the mountain but we

[22:07] to acknowledge those who have mapped out the route beforehand. And there were plenty of people that helped map the route to this solution So the equations were written down in the middle of the 19th century First big breakthrough was probably around 1934 This guy LeRay So what did he do So LeRay changed

[22:24] the equation slightly. He took the Navier-Stokes equation and he multiplied it by a smooth function. It's like a type of smooth velocity, like a tesla velocity. So he multiplies the

[22:36] equation by a smooth function, no wild gradients, nothing like that, and he averages over it. and he uses some tricks of calculus, this allows him to move some of the gradient operators away from your velocity field onto this new

[22:51] function and so things that might have blown up no longer blow up because they're acting on this smooth guy and he's able to show that the solutions of this sort of smoothed out system will exist for any amount of time

[23:05] okay and that's called a weak solution but of course he's not really dealing with the full, now that he's so excited, he's dealing with a smooth version of it. But he showed that they exist. So that was the first big result, I would say.

[23:17] And then there were sort of various other results that came along. So I think in the early 80s, there's the CKN collaboration. They showed that if these blow-ups exist, if they do happen, then they're necessarily

[23:29] very sparse in the fluid. So you're not going to get a whole bunch of them in one place. That ain't happening. They're going to be very sparse. There were other people, I think the ESF collaboration, who showed that if you impose some extra constraints,

[23:42] so same things aren't blowing up too fast or certain quantities don't grow, adding these additional agreements proves that things can't blow up. But again, it's not quite the same system. One of the really big breakers was in the mid-2010s, your friend Kenneth Powell, of course.

[23:58] What did he do? Well, he took the Navier-Stokes equation. If we go back to the equation, I've got this one. and he essentially just looked at the equation but he replaced this term so we know this term

[24:10] is the naughty term okay so so what tar did was he replaced this term with like a dress version of it so the dressing allows him to sort of scale various parts it allows him to block out certain

[24:23] sort of you know momentum modes or manipulate them in a certain way so he replaced this term with this term which he could manipulate. He shared a lot of properties with that guy. And it allowed him to sort of figure out a way

[24:36] to sort of transfer energy to smaller and smaller scales and engineer a blow-up. But he wasn't dealing with true Navier-Stokes. He was dealing with something which was very Navier-Stokes-like,

[24:48] but it had this dressed contribution that he'd introduced. So it's not quite Navier-Stokes, but shows that a blow-up happened. And then very recently, we have the CMZ collaboration who did something really quite remarkable.

[25:03] So they weren't working with Navier-Stokes. They were working with the Euler equation. Now, the Euler equation ignores the viscosity. Right. Okay. Now, that's important because we know that the viscosity likes to smooth things out.

[25:18] Okay. So you're more likely to get a blow-off if you ignore the viscosity. I think everyone would agree that. So what did they do? Well, they showed that you can have a configuration where you have like almost a clockwork of vortices.

[25:31] So you have one vortex, another one, another one, and they're kind of hierarchical in size. And the bigger vortices can sort of amplify the effect of their smaller neighbors. And you have an infinite number of these. And then even though the energy of the system is finite, this infinite clockwork can generate a blow-up.

[25:47] but the problem with what they did was their force was not quite smooth it was highly regular but it wasn't completely smooth

[25:59] in a mathematical sense it wasn't quite a solution now, fast forward to 2025 and 2026 and you have a professor at NYU bookmaster and an employee at Anthropic

[26:13] I'll tell you, I'm sure I'm pronouncing that wrongly and they basically took the CMZ idea and they were tackling the Euler problem, so again, no viscosity, and they were able to find a situation

[26:25] with this nice sort of clockwork, this cascade of vortices, but with smooth forces. Okay. So like a real sort of, on the brink of really getting towards an RBA spokes now.

[26:37] Now what, this is where things get a little bit dodgy, maybe. Okay. So OpenAI got wind, the DC guys were on the brink that something was happening.

[26:49] Whether they knew about them directly, but they heard that some of the Millennium problems were perhaps on the brink of being solved. So they went into overdrive. Right, let's attack the Millennium problems. They zoomed in on Navier-Stokes

[27:01] and actually very quickly they solved the Euler problem, not just for a smooth force but for vanishing force, actually. And then of course they switched to

[27:13] Navier-Stokes, the full Navier-Stokes with the viscosity included and they and then they solved that too within 88 hours. Now there is some debate and you know that the NYU prof and the anthropic guys that they were

[27:29] they've been using the sort of OpenAI codecs to sort of interact with some of their ideas and maybe the OpenAI systems were aware of that maybe directly or indirectly and that could have influenced

[27:41] at least their solution to the Euler problem. That's, I mean, maybe, maybe not. But what I will say, and I'm not one to defend O2AI,

[27:54] Salomon is not my favourite person, but I do think that the solution to Navier-Stokes rather than the Euler problem without the viscosity is actually very different to the way people were

[28:07] attacking this Euler problem. it is quite different one has these pulses the Navier space has these pulses the other has this sort of cascade this clockwork of water it's very different so I certainly have to acknowledge

[28:19] that it did apparently get quite unpleasant bookmaster apparently there was communication between him and OpenAI OpenAI offered that he could potentially be an author on the papers

[28:31] but they didn't want the Alpurgia involved because of course they had this rivalry around tropic I'm really impressed with the way bookmaster just looked at his guns and said, no, I've worked with him, this is not right.

[28:43] Yeah, and we are where we are. There's a whole bunch of links below. Firstly, to a continuation of this discussion with Tony, where he talks about how he feels about AI being used for mathematics

[28:55] and some of the philosophical questions that come from it. The hard mathematical problems, the reason that they've been chosen is because the journey to solving them is the important thing.

[29:08] There'll also be a link to Tony's book, more videos with Tony, and also links to previous videos we've done about Navier-Stokes and Reynolds numbers. We don't even know if a solution's going to come out. The full Navier-Stokes equations, as written down on our piece of paper.

[29:23] Hang on, which way are we now, though? We're talking this, yeah. So those should be exactly the ones that are written down. The little guy. The little guy and the big guy. What did you call it? You called it the, uh... The wild card.

[29:35] Have you seen Odyssey, Brady? like they've got this whirlpool haven't they right so they replace that whirlpool with the wild core right what's going to happen