For centuries, mathematicians have wondered whether Euler's fluid equations would collapse or be "exploded" in some cases. A new approach to machine learning has convinced researchers that this "blasting" is coming.
The author | Jordana Cepelewicz
Compiled | Qian Lei and Ailleurs
Editor| Chen Caixian
For more than 250 years, mathematicians have tried to "blast" some of the most important equations in physics, such as Euler's equations, which describe fluid flow. If they succeed, they will find that in some case the equation will explode—for example, a vortex that rotates infinitely fast, or a current that suddenly stops and flows suddenly, or an electron that passes by at an infinitely fast speed. Beyond this flashpoint — the "singularity" — the equation will no longer have a solution. These equations would not even be able to describe the ideal situation of the world, and mathematicians have reason to wonder whether models of these fluid behaviors are reliable.
The singularity is as slippery and elusive as the fluid it is describing. To find out, mathematicians typically feed the equations that control fluid flow into computers and then do digital simulations. They start with an initial set of conditions and then observe until the value of a certain quantity—such as velocity, or eddy—begins to grow wildly, seemingly moving in the direction of an explosion.
But the computer can't find the singularity with certainty, for the simple reason that the computer can't handle infinite values. If a singularity exists, the computer model may approach the point where the equation was blasted, but never get the singularity directly. In fact, when probed with more powerful computational methods, the obvious singularity has disappeared.
But this approximation of singularities is still important. With approximations, mathematicians can use a technique called computer-aided proof to prove that there is indeed a singularity nearby. There have been simplified one-dimensional versions of studies before.
Earlier this year, a team of mathematicians and geoscientists discovered a completely new approach to approximate singularities — they utilized deep learning methods to be able to directly observe singularities. The team also used this approach to look for singularities that traditional methods could not find, hoping to prove that these equations are not as infallible as they seem.
Address: https://arxiv.org/pdf/2201.06780v2.pdf
In the study, Yongji Wang et al. developed a new numerical framework that uses physics-informed neural networks (PINNs) to find smooth, self-similar solutions to Boussinesq's equations. The solution corresponds to the asymptotic self-similarity curve of the three-dimensional Euler equation with a cylindrical boundary. In particular, the solution is an accurate description of the scene of the three-dimensional Euler equation Luo-Hou blast. The solution is the first true multidimensional smooth backwards self-similar curve of the hydrodynamic equation. The numerical framework is robust and easy to apply to other equations.
This paper investigates the finite-time blasting problems of the two-dimensional Boussinesq equation and the Euler equation with three-dimensional band boundary, which are of great significance in the field of mathematical fluid mechanics. Yongji Wang et al. used a novel numerical method to construct a smooth backward self-similar solution to Boussinesq's equation using a physical information neural network. This solution itself could become the basis for future bursting of Euler equations for computer-aided proofs of two-dimensional Boussinesq and three-dimensional band boundaries.
The study sparked a race to blast fluid equations: deep learning teams on one side, and mathematicians who have been using more mature techniques for years. Whoever might win the race — if anyone could actually reach the finish line — the results showed that neural networks could help people find new solutions to many different problems.
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Disappearing burst crack
Leonhard Euler proposed Euler's equation in 1757, which describes the motion of an ideal, incompressible fluid—a fluid that has no viscosity, no internal friction, and cannot be compressed to smaller volumes. (Many of the fluids found in nature are likewise sticky, and their model is the Navier-Stokes equation; the Blasting Navier-Stokes equation will receive the Clay Institute for Mathematics' $1 million Millennium Award.) Given the velocity of each particle in a fluid at a certain starting point, Euler's equations should be able to predict how the fluid will flow at any one time.
But mathematicians wonder if, in some cases — even if they don't seem problematic at first — the equations will end up in trouble. (We have reason to suspect that this may be true: the ideal fluid they simulated bore no resemblance to a true fluid with only the slightest viscosity.) The formation of singularities in Euler's equations can explain this divergence. ）
In 2013, two mathematicians came up with such a vision. Since the dynamics of a complete three-dimensional fluid flow can become incredibly complex, California Institute of Technology mathematician Thomas Hou and Guo Luo of Hang Seng University in Hong Kong argue that flow obeys some kind of symmetry.
In their simulation, the fluid rotates inside a cylindrical cup. The liquid in the upper half of the cup rotates clockwise, while the liquid in the lower half rotates counterclockwise. The opposite flow of water forms other complex up-and-down circulation of water. Soon, at the intersection of two opposing currents of water on the border, the vortex of the fluid exploded.
Image source Merrill Sherman/Quanta Magazine
Although this proof provides strong evidence of the existence of a singularity, without evidence, it is impossible to determine that it is a singularity. Before Hou and Luo's proof, many simulations suggested potential singularities, but later when tested on a more powerful computer, most of them disappeared. Vladimir Sverak, a mathematician at the University of Minnesota, said: "You think there is a singularity, and then you put it on a larger computer with better resolution, and somehow the singularity that you thought existed is gone." ”
That's because these solutions can be susceptible to seemingly insignificant errors that accumulate with each time step in the simulation. Charlie Fefferman, a mathematician at Princeton University, said: "Simulating Euler's equations on a computer is a delicate art because Euler's equations are very sensitive to small and small errors of 38 decimal places after the solution. ”
Still, Hou and Luo's approximate solution to the singularity has so far withstood all the tests and has encouraged many people to do the research. Sverak said: "This is the best solution for singularity formation, and many people, including myself, believe that what is obtained this time is a real singularity." ”
To fully prove that Euler's equations have been blasted, mathematicians need to prove that there is a real singularity nearby given an approximate singularity. They can re-describe this statement in precise mathematical terms, such as the existence of a real solution in a region close enough to the approximation, which can be proved correct if certain properties can be verified. Verifying these features requires a computer: this time it is necessary to perform a series of calculations (including approximate solutions) and carefully control the errors that may accumulate in the process.
Hou and his graduate student, Jiajie Chen, have been working on computer-aided proofs for several years. They improved the approximation solution starting in 2013 and are now using this approximation as the basis for their new proof. They also showed that this general strategy also applies to problems that are easier to solve than Euler's equations.
Now, another group of people has joined the hunt. They found an approximation using a completely different approach, which is very similar to the results of Hou and Luo. They are now writing their own computer-aided proofs. But in order to get an approximation, they first need to move to a new form of deep learning.
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PINN: It started with glacier research
New research on blasting in Euler's equations began in an unexpected field where geophysicists studied the dynamics of antarctic ice sheets. Their research called for the use of a deep learning approach that later proved useful in more theoretical contexts.
Mathematician Tristan Buckmaster, who is currently a visiting scholar at the Institute for Advanced Study in Princeton, discovered that this new approach was purely accidental. Last year, Charlie Cowen Breen, an undergraduate student in his department, asked him to sign a project that had been doing dynamics research on the Antarctic ice sheet under the direction of Princeton geophysicist Ching-Yao Lai. They tried to speculate on the viscosity of the ice through satellite imagery and other observations and predict its future flow. They did this by using a deep learning method never seen before, a neural network based on physical information (PINN).
Unlike traditional neural networks, which require training on large amounts of data to make predictions, PINNs must also satisfy a set of potential physical constraints, including laws of motion, conservation of energy, thermodynamics, and so on, as well as any other physical constraints that scientists need to introduce in order to solve a particular problem.
Image source NASA Earth Observatory
Introducing physical factors into neural networks serves several purposes. On the one hand, such neural networks are capable of answering questions with little to no available data. PINNs, on the other hand, are able to infer unknown parameters from the original equations. Yongji Wang, a postdoctoral researcher in Lai's lab and one of the co-authors of the new paper, points out that in many physical problems, "we roughly know what the equations should look like, but we don't know what the coefficients of 'some' terms should be." This is the case with the parameters that Lai and Cowen Breen are trying to determine.
George Karniadakis, an applied mathematician at Brown University who developed the first PINNs in 2017, proposed and named them "hidden fluid mechanics."
The request of the student Cowen-Breen provoked The Buckmaster's thoughts. Hou, Luo, and Chen et al. have undergone a long and difficult advance in solving the classical solution of Euler's system of equations at the cylindrical interface. But because of their dependence on time, they can only get very close and cannot reach the singularity: as they get closer and closer to something that might look like infinity, the computer's calculations will become so unreliable that they will not be able to really see the point of the blast itself.
But Euler's equations can be represented by another set of equations, and with a clever technique, the effects of time can be excluded. Hou and Luo's (2013) findings are remarkable, not only because they identified a very precise approximation solution, but also because the solution they found seemed to have a special "self-similar" structure. This means that no matter how long time goes forward, the model's solution follows a certain pattern: its later shapes look very similar to the original shape, just larger.
This feature means that mathematicians can focus on a certain time before the singularity appears. If they zoom in on that snapshot at the right speed—it's as if they're constantly making adjustments under a microscope to zoom in to see it—they can simulate what happens after that until they reach the singularity itself. At the same time, if they rescale in this way, there will actually be no serious errors in this new system, avoiding the problem of dealing with infinite values. "It's just approaching a good limit," Fefferman says, which represents the occurrence of blasting in a time-dependent version of the equation.
"It's easier to model these [rescaled] functions, so if you can describe a singularity with a self-similar function, that would be a big advantage," Sverak says. ”
From left to right: mathematicians Tristan Buckmaster and Javier Gómez Serrano, geophysicists Cheng Yao Lai and Yongji Wang. They collaborated to study the blasting of Euler's equation using a physically-based neural network.
The problem is that for it to work, mathematicians don't just ask to solve equations for usual parameters such as velocity and eddy (writing these equations using self-similar coordinates), but the equations themselves have an unknown parameter: variables that control magnification. This value must be just right to ensure that the solution of the equation is consistent with the burst crack in the initial problem.
Mathematicians must solve these equations both forward and backward—a task that is difficult, if not impossible, to achieve with traditional methods. Finding these solutions is exactly what PINNs are designed for.
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The path of explosive cracking
Looking back, Buckmaster says, developing PINN "seems obvious."
Buckmaster, Lai, Wang, and Javier Gómez-Serrano (a mathematician at Brown University and the University of Barcelona) collaborated to create a set of physical constraints to help guide PINNs, which include conditions related to symmetry and other properties, as well as the equations they want to solve. They used a set of two-dimensional equations rewritten using self-similar coordinates that are equivalent to three-dimensional Euler equations at points close to the boundaries of the cylinder.
They then trained the neural network to look for solutions that meet these constraints—as well as self-similar parameters. "This approach is very flexible, and as long as the correct constraints are applied, you can always find a solution." Lai said. In fact, the team demonstrated this flexibility by testing the method on other issues.
The answers provided by the team look a lot like the solutions proposed by Hou and Luo (2013). But mathematicians hope that the approximation they give will describe in more detail what is happening, because this is the first time that a self-similar solution to the problem has been directly calculated. "The new findings illustrate more precisely how singularities are formed," Sverak says, i.e. how certain values will reach burst points and how equations will collapse.
Buckmaster notes: "Without neural networks, it's hard to prove that you're really capturing the essence of the singularity. It's clear that the method used in this study is much easier than the traditional approach. ”
Gómez-Serrano agrees, saying: "This will become a standard tool at people's disposal in the future".
PINNs once again reveal what Karniadakis calls "hidden fluid mechanics," only this time, they have made progress on more theoretical problems with PINNs. Karniadakis said: "I haven't seen anyone do this with PINNs. ”
That's not the only reason mathematicians get excited. PINNs may also be used to find another kind of singularity that is almost impossible to find with traditional numerical methods. These "unstable" singularities may be the only ones present in some hydrodynamic models, including Euler's equations without cylindrical boundaries (which are much more complex to solve) and Navier-Stokes equations. "Unstable singularities do exist. So why not find them? Princeton mathematician Peter Constantin once said.
But even for stable singularities that can be handled with classical methods, PINN's solution for Euler's equations with cylindrical boundaries "is quantitative and precise, and can also become more rigorous." Now there is a roadmap to proof. It's going to take a lot of work and a lot of skills. I think it also needs some creativity. But I don't think it takes any talent. I think it's doable. Fefferman said.
Buckmaster's team is now in a race with Hou and Chen to see who can get to the finish line first. Hou and Chen are one step ahead on this track: According to Hou, they have made substantial progress in improving the approximation and completing the proofs over the past few years, and he suspects that Buckmaster and his colleagues will have to improve the approximation solution in order to get their own proof. He believes that the margin of error for existing approximation solutions is already small.
Still, many experts hope that 250 years of exploration of cracking euler's equations will come to an end. Sverak said: "Conceptually, I think ... All the important parts are in place, but the details are still difficult to determine. ”
Reference Links:
https://www.quantamagazine.org/deep-learning-poised-to-blow-up-famed-fluid-equations-20220412/
https://arxiv.org/pdf/2201.06780v2.pdf
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