We've talked on this podcast before about the equations that mathematicians and physicists use to describe fluids in motion. They're called the Navier-Stokes equations, after the two physicists who devised them 200 years ago. They're great as equations go. They're powerful and accurately describe how fluids should behave. Water flowing through a pipe, waves on the ocean, air passing over an airplane wing. these are all real-world solutions to those Navier-Stokes equations, but that doesn't mean that the math behind them is easy to handle. Welcome to the Quanta Podcast, where we explore the frontiers of fundamental science and math. I'm Samir Patel, editor-in-chief of Quanta Magazine. Finding hidden depths in the math of the Navier-Stokes equations was the subject of a recent story by Quanta staff writer Charlie Wood. called Using AI, Mathematicians Find Hidden Glitches in Fluid Equations. And Charlie's on the show to talk us through it. Welcome back, Charlie. Hey, Samir. I'm delighted to be here. So what's the big idea? So this story has some physics energy to it, which is what got me into it as a physics reporter. But really, as you mentioned, it's a pure math story. So the physics energy comes from the fact that we have these incredible equations, the Navier-Sokes equations, and they're just as rich and they're just as beautiful as the equations that describe quantum mechanics, So the equation describes space-time and general relativity, but they describe fluids, so whirlpools, water flowing, all that good stuff. So that's the physics side. Now, these equations, being equations, are also pure math objects. And partially because they're so famous and so interesting, mathematicians are also just motivated out of pure curiosity to understand them at the mathematical level. So the big idea here is mathematicians want to answer a simple yes or no question about these mathematical objects. That is, do all of their solutions make sense? Are they physical at all places and all times? Do they describe liquids that always behave in a logical way or not? The Navier-Stokes equations are very successful in general with describing real phenomenon in fluids. But what we're looking at here is probing the limits of where they might actually break down and not be able to describe something in the real world. Yeah, exactly. Got it. Okay, so when you're describing the Navier-Stokes equations to a non-physicist, to a non-mathematician, how do you typically describe them? So they are the fluid version of another famous and perhaps more familiar equation, Newton's second law of motion. That's F equals MA, force equals mass times acceleration. F equals MA tells you about what an object is doing in response to a certain force. And it has lots of solutions. It describes a book or a laptop sitting on a table and not doing anything. It describes a rock falling off of a building. It describes a cannonball's parabolic trajectory through space, right? So these are all different solutions to ethical Zembe. And the Navier-Stokes equations are like that, but for fluid. They're actually very similar. Force applied to different points of the fluid and what happens as a result. And so the different solutions to the Navier-Stokes equations are similar. It could be water in a fish tank just sitting there doing nothing. It could be a whirlpool, water slowing down a river. So at a mathematical level, these are both differential equations. Maybe a little bit of a scary word, but that just means an equation that has a derivative in it, which is just a rate. It's something changing in time or in space. And what's interesting about differential equations is that their solutions are not numbers. Their solutions are curves, functions, equations themselves. So the equations are very powerful. Now, they're used a lot in modeling, right? So if you want to model the flow of air over an airplane wing, those equations are actually actively being used in that kind of computer model, right? But is there a reason that mathematicians are looking for strange behavior of these equations? Right. And again, this is not a physics problem. This is a math problem. Yeah. We are concerned with these mathematical solutions. And those, as I mentioned, are their functions, their equations, their curves. and there's an infinite number of them because you could put water into an infinite number of situations. And all of those are solutions. And so those are functions and the function can do whatever it wants. It does not have to make physical sense. So take a familiar function that we might have encountered in high school, 1 over x. What is it doing at 0? Well, 1 over 0, it's not a number, it's undefined. And we might say it's like infinity and that's totally fine for the function. But if I want to tell you that this function describes the position of my laptop on a table, you should doubt me. Because where is it when time equals zero? That doesn't have a physical interpretation. So there's this logical possibility that of all those different things that water could be doing according to the Navier-Sokes equations, that they might have these undefined places that have no physical interpretation. The title of the story called it a glitch. Now, we'll talk about what that means in a second, but reality doesn't glitch out. Water doesn't just instantly change direction or create a tiny column of water that goes up to the sky. That's not the way things behave in the real world. But on a purely mathematical level, such things are possible when you're dealing with complex math. So what are some of the mathematical glitches? What do we call them? What are these things that mathematicians are looking for? They have a few names. Sometimes they're called blow-ups. Sometimes they're called singularities. I think of them as glitches from the physics side. But these are all basically the same thing. They're a place where the solution becomes undefined and really infinite. And again, infinity has no physical meaning. No physical measurement can ever return infinite number of things, infinite speed, infinite location. That's kind of nonsense. But this is fine on the math side. Even if it can happen in equations describing real physical stuff mathematical equations end up with places where things like 1 over 0 happen all the time 1 over 0 is undefined but if you graph the function you'll see that the closer it gets to the point where you're trying to divide 1 by 0, the more the graph goes up and up and up, zooming toward infinity. It blows up, Mathesians say. So one thing that they call these glitches is blow-ups. So is there a specific kind of blow-up that the mathematicians are most interested in? In the context of Navier-Stokes, basically any blowup. And they look for blowups in density and velocity and vorticity. And any time anything becomes infinite, that's a blowup. So mathematicians are looking for these blowups in the Navier-Stokes equations. How are they doing that? So this is where Navier-Stokes is very different from Epicles of May, which has tons of solutions that are easy enough for high school students to write down and solve. Fluids are changing in lots of different ways all throughout space. So we really can't write down a simple function that tells you what's happening everywhere all the time. So people simulate. These are like digital experiments that you run on your computer. This is like, in the F-E-C-O-S-A case, taking a digital cannonball with gravity programmed into your computer and everything, firing it off, and then you let the digital cannonball rise and fall. So you can do this with fluid, too. You set it up in a container with a force and whatever, and you play it forward kind of frame by frame, and you build up a video of what's happening over time. With the actual equations being the thing that is dictating the behavior of that simulated fluid. Telling you what's happening one moment after another. Got it. And if you can get something to glitch out, to blow up in that simulation, that indicates something is happening with the math. You can go and see what that is. Exactly. And that's rather subtle because computers can't represent infinity in their memory. They glitch. They glitch. So eventually your computer, you're going to see velocity, for example, getting faster and faster and faster and faster. And then your computer crashes. That's a really important sign, though. And it tells you you should go back to your math and look carefully at what's happening at that point. But there's a challenge or a problem with this approach, which is that computers are imprecise. They are digital. We have to pixelate our fluid. We have to step it forward in time. And we have to add a pixelation in time. And the Navier-Stokes equations are describing a perfectly smooth fluid, no molecules, and a perfectly smooth time. And so our digital experiment has these digital artifacts that separate it just a tiny bit from the actual continuous underlying mathematical object that we want to understand. Yeah, that's like a third representation of this, right? We have the reality, physical reality of it. We have the pure continuous math version and then the digital simulation of that, which is just a little grainy. Exactly. And so if you see a solution or a candidate, really, as your computer simulation starts to blow up, then you might be unsure whether the underlying continuous version truly blows up at the same point in the same way. This brings us to a really important distinction in two different types of singularities that's crucial to understanding the story, stable versus unstable. A stable singularity is one in which, as the fluid is evolving towards this blowup, if you poke it a little bit, it does not change its fate. It continues to evolve in the same way and blows up. A durable blowup. A durable blowup, exactly. Durable enough to withstand a little bit of pixelation in our computer simulation. Got it. In contrast, an unstable singularity is a fluid that has unlimited sensitivity. Any change, no matter how small, will change where it's going and will change its behavior. So if it were on the path to a blowup and you may move one point, one tiny, tiny distance to the left. Any little change. Any little change, no blowup. And so those are the singularities that it is not possible to find using a computer simulation. Okay. And inconveniently, those are the solutions of the singularities that mathematicians, They have a hunch that if Navier-Stokes equations have blow-ups, they're probably going to be of this type for a few different reasons. They tend to happen very quickly before the fluid has a chance to rearrange and disperse all of the energy and stuff like that. So there's a lot of interest in being able to find these. Okay, so just to recap a little bit. We have this idea of blow-ups in the Navier-Stokes equations. We're divorcing this from reality, even though they're used to describe reality. We have the potential for these blow-ups. the blow-ups are where something goes off in the way that that equation might reflect reality. Something goes to infinity. Some of these are resistant to perturbation. That's a very nice phrase, right? Yes. That if you poke and prod, that singularity as it's starting to form, it will continue to form. Yeah. And those are easily simulated on a computer. Not easily. Okay. It takes years and years and years to do this. Those can be simulated on a computer. Yes, and years to prove they are, in fact. That's a tremendous amount of work. And then at a purely mathematical level, because these almost can't be by definition simulated on a computer, are these unstable blowups that is a very extremely, almost infinitely precise set of conditions to create infinity in the equations. So precise the computer does not have the precision to recreate them. Okay. Are there stable blowups that we know of emerging from the Navier-Stokes equations? No. Okay. If anyone finds one, they will be entitled to a million-dollar prize from the Clay Institute for Mathematics. There are no known blowups in the Navier-Stokes equations. No known blowups in Navier-Stokes. If you can prove that one exists or prove that none exist, there's a million-dollar prize for that. Okay. So this is a big motivation for people to work in this field. Right, right. That's not bad. For mathematicians, can you explain why the unstable blowup is more interesting? Or is that not the right word to describe the distinction between the two? Certainly, different mathematicians will have different opinions on this, I think. But the sense that I got, again, from a physics point of view, which I don't know if they'd all sign off on, is that the Navier-Stokes fluids almost don't want to blow up if I can anthropomorphize them a little bit. They tend to kind of skitter away as soon as, I mean, it makes sense. If something moving very very fast or has a ton of energy then that wants to disperse And it can disperse in a few ways It can move in a different direction Or in the Navier equations there something called viscosity which is kind of internal friction We sometimes think of it as how thick the fluid is. Viscosity takes that growing energy and spreads it out. It dissipates it throughout the fluid. So viscosity works against the possibility of a blowup. And this is a feature, not a bug, because it's one of the things that makes these so effective for actually describing reality or modeling fluids. Yeah, yeah. You don't want your fluids to blow up. I mean, water doesn't blow up. And part of the reason it doesn't do that is because it's got viscosity. Yeah, sure. We like viscosity. Yeah. But unstable singularities can kind of spring up out of nowhere and happen in a certain sense faster than other ones. And so they can happen before viscosity even has a chance to start to spread things out. And so they can kind of overwhelm the tendency of viscosity to slow things down. So that's one motivation that people are interested in. But these blowups, stable or unstable, have yet to be found in Navier-Stokes equations. So how are they going about looking for them? They're working in simpler contexts and also contexts where the abstract fluids being described are more likely to blow up. So I mentioned that one way you can avoid a blowup is to have the fluid sort of spread out in space. So you might constrain it. You might put it between two flat glass plates and make it move in two dimensions. Or you might put it in a container with a boundary and have a fight against this boundary. Or you might take viscosity out and have this very slippery fluid that is easy to blow up. And we do know of stable blowups in those contexts. The most famous one and kind of the landmarks or frontier of this field. The candidate was found in 2013 by Thomas Ho at the California Institute of Technology and Go Lo, now at the Hang Seng University of Hong Kong. They were doing a pixelated simulation of fluid spinning against a cylindrical can. And the symmetry of the can is important for the math. And in their simulation, they used a different, older set of equations called the Euler equations, which are like the Navier-Stokes equations, but they have no viscosity. So they were able to do the simulation, and they found signs of a blowup. Then, over the next 10 years, Thomas Ho and his graduate student, Jiajie Chen, who's now at the University of Chicago, were able to rigorously prove that this blowup was real, under these simplified conditions. That's stable. So that is stable because they were able to find it with a simulated pixelation. Okay. Yeah. So they've constrained it physically and they've taken variables out of the equation, viscosity out. So this is a hint that these kinds of blowups could exist in a more complex scenario. That's right. This is a somewhat contrived setup, but if you can get it to work here, why not get it to work in the full situation? So that's a stable blowup in a simplified system. Have mathematicians found an unstable blowup in a simplified system? Or to put it another way, these systems and equations that you're simulating, how simplified do you have to make them to create an unstable blowup? There have been one or two examples of unstable blowups, but in incredibly simplified one-dimensional systems that happen to be related to a different equation, but that is kind of a lucky coincidence. But recently, there's been a pretty significant breakthrough in the ability to look for unstable blowups, and this has developed from a whole new technique for finding blowups that involves artificial intelligence. Last fall, they unveiled five or ten new unstable candidate blowups in three fluid setups. And when we say fluid setups, these are physical situations. One of them is the same cylindrical can where Ho and Lo found their candidate back in 2013. And in the same setup, the new research has uncovered additional unstable blowups that you can get by setting the fluid spinning in slightly different ways. Okay. So using AI, this is the news, right? Using AI, researchers have gotten unstable blowups in more complex fluid setups. So how did they do this with AI? This is the key idea here, right? Is that they've been developing another general technique that can stand alongside simulation as a second way of looking for blowups. The way that this works is rather than build up the movie of the fluid frame by frame, this instead is more like using AI to go after the whole solution directly. And that's why it works is because you aren't stepping it forward in time where errors can accumulate and the sensitivity of the unstable fluid can knock you off your trajectory. You kind of go for the whole thing all at once without using time, without moving forward frame by frame. So a few years ago, Tristan Buckmaster and Ching-La Wai, then at Princeton University, realized that physicists were already developing neural networks to do this sort of task. In the end, these neural networks end up approximating a function. You enter an input and get an output. That's what a function does. So you can train a neural network to approximate a function that solves a differential equation, like Navier-Stokes. That's the vision. And yet Buckmaster and Wai came across some physicists who were using neural networks, specifically a type called a physically informed neural network, to solve differential equations. And the reason why it's called physically informed is because differential equations are common in physics, and so they're specialized for this task. So specialized AI for examining particularly complex formulas. Examining a differential equation and adjusting itself, adjusting the numbers in the neural network until the neural network itself is a solution, is a function that solves the complicated equation useful in physics. So they were able, through this process, to take some of these fluid setups and tweak them to the point that they did generate blow-ups. Right So they been doing this for a number of years Buckmester and Wai along with their collaborators Yongji Wang and Javier Gomez Serrano have been taking these pins and refining them and tailoring them exactly to the specific equations they want to solve They done very technical work over a number of years We actually have a previous article that also covered their earlier results But most recently the headline result is that they were able to get these neural networks to contort themselves into forms that seemed like they have these singular behaviors Specifically unstable behaviors Because again they not stepping the equations forward in time They are getting the neural network to adopt a shape that satisfies the equation all at once. Tristan Buckmaster described this as like trying to balance a pencil on its tip. Yeah. So the million-dollar prize is for finding blowups in the Navier-Stokes equations. The previous results had shown that stable blowups were possible in simplified systems. And now we've shown that unstable blowups are possible because of this sort of innovation in the way that they're using the AI. So we're a step closer to finding blowups of any sort in the full Navier-Stokes equations. So they're kind of tackling the technical problems one by one. Does it work in 2 and 3D fluids? Yes. Does it work in fluids with something like viscosity? Yes. And the hope is that someday, and this is totally not guaranteed, but their ambition is to someday get it to work in the full Navier-Stokes equations, which are 3D, no boundary, and has viscosity. So they've got a few more pieces of this puzzle, but we're still a ways away from putting it all together in the way that would solve this extremely challenging math problem to find this needle in a field of haystacks. Yeah, infinite haystacks. And I should mention that the next major prize for this field, the next news to watch out for, is called the boundary-free Euler equations. And so this is that spinning cylindrical solution from home and go. Can you take the cylinder out? No viscosity, but infinitely expansive 3D fluid. Can you find a singularity in that situation? And there's a number of teams racing for this prize. The work that I just described is one of those teams. And there are other people using pencil and paper methods and looking for solutions that are stable in some ways, unstable in others. But I think with a handful of years, it's plausible that we could see a solution to that. And then Navier Soaks is significantly harder than that. And so who knows when that would happen? Right. Well, this is, it's cool to me how there's a whole bunch of teams that are trying to raced toward this goal of breaking some very august and venerated and quite well-established equations for describing the world. Yeah, that's true. It's kind of beautiful to see the passion that these pure questions inspire in people. This really is the abstract love of mathematical truth, just for the love of the game. Yeah. I've really enjoyed talking about this with you, Charlie. Thank you so much for coming on the show. This has been great, Samir. Thanks for having me. we like to close every episode with a recommendation so what's exciting your imagination this week well since we've been thinking about fluids today i'll recommend what is probably my favorite non-fiction book of all time it's called the emerald mile and it's a rip-roaring account of these hooligan guides who try to set a speed record going down the grand canyon decades ago and this normally takes people weeks to do my dad and i did this a few years ago had a great time and they try to do it in a few days it's an incredible epic tale outdoor adventure but even better than that aspect of the book, which is only like a third of the book. The other two-thirds is to understand the wild conditions that led to this speedrun being possible. You have to understand the history of the dams in the river and why they were put where they are and the history of guiding culture in the river and humans in the river. And so really, most of the book is the history of humans in the Grand Canyon culminating in this epic attempt. Thanks for that, Charlie. I'm going to check that out. Sure. You're welcome. Also on Qantas this week, you can read another story about the use of AI, this time to map the brain in ever greater detail, like going from regions to neighborhoods. And another Matt story about differential equations and how we know whether they're well-behaved enough to model reality. I don't know if you've noticed, but we often describe some real-world image of water or air as a quote-unquote solution to the Navier-Stokes equations, and there's a reason for that. The equations describe what is physically possible within the fluid model that they establish, So a flow in the real world that behaves the way the model predicts must satisfy the equations in some particular way. And that's what makes it a solution. So with that in mind, we're going to leave you today with the sound of a solution to the Navier-Stokes equations. Thank you. Transcription by CastingWords Our theme music is from APM Music. If you have any questions or comments for us, please email us at quanta at simonsfoundation.org. Thanks for listening. From PRX.
