Euler’s equations, proposed in 1752, and those of Navier-Stokes – between 1822 and 1842 – are fundamental tools to describe, in mathematical terms, the behavior of incompressible fluids – that is, those that cannot be compressed, such as water. –. They allow us to understand natural phenomena such as the flow of rivers or the breaking of waves. However, solving these equations – even with the help of powerful computers – is very difficult and expensive. Even 250 years after being written, they represent a mathematical mystery.

One of the fundamental questions about them is whether, starting from soft initial conditions – for example, if the sea is calm – a singularity – a tsunami – can be triggered in the equations after a finite period of time. In mathematical terms, the singularity occurs when the variables that model the physical phenomenon, such as speed or pressure, go from having finite values to being infinitely large, in a finite time. This is one of the so-called problems of the millennium – that of the Navier-Stokes equations – whose solution is rewarded with a million dollars.

Although there have been partial advances (like this one and this one), so far, all attempts to resolve this issue have failed. One of the reasons is that singularities are not understood well enough. For example, we don’t know how to predict how quickly a whirlwind may form or where it will occur.

With a computer we can improve our understanding of these objects, through numerical experiments that closely approximate certain solutions of the equation, candidates for producing singularities, that is, for taking on infinite values, in a finite time. Over the years, the community has identified several such numerical solutions. And, as computing power has grown, these candidates have been refuted or validated and complemented by others.

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In 2014, Guo Luo (researcher at Hang Seng University in Hong Kong) and Thomas Y. Hou (Caltech University) produced a simulation of a fluid inside a cylinder, which, under certain starting conditions, appeared to give place to a singularity.

Representation of the Luo and Hou simulation, in which the fluid moves inside a cylinderJavier Gómez

Luo and Hou claimed that their uniqueness was of a specific type, locally called self-similar. In this type of solutions, as we approach the moment of the singularity, if we zoom to the appropriate scale, we observe the same solution again, as if it were a fractal.

Locally self-similar singularities have a fractal appearance like the one seen in the image. Image: We are not clear if it is possible to use this image or how to cite the source. Maybe you can look for another similar one.

But, despite the promise of the approach, its simulations took months to calculate and were difficult to replicate. Furthermore, they only offered solutions of a specific type, called stable ones, which, according to the general opinion of the mathematical community, would not allow the Navier-Stokes problem to be solved. Recently, almost 10 years later, an interdisciplinary group of researchers formed by a Spanish mathematician and an Australian-British mathematician, and two geophysicists – Chinese and Taiwanese – have confirmed Luo and Hou’s proposal, finding self-similar solutions for several types of equations of the fluids. Some of them are of an unstable type – that is, like those that are expected to solve the millennium problem. This new method allows us to understand where and how the singularity is formed and the shape of the solution at any time before the explosion, even an instant before.

To do this, they have made use of machine learning techniques in applied mathematics. Specifically, the researchers start from an approximate solution that is refined thanks to a neural network, assessing in each iteration how well the approximation satisfies the equations, to improve the solution. For example, if by increasing the value of the solution at a specific point, the error made when approximating the equation becomes smaller, the neural network will try to find a new candidate in that direction.

This is the first time that machine learning has been used to solve fluid mechanics problems, which, for the group of scientists, has made it possible to better approximate the solution of the studied equation, also using much fewer computational resources. Thus, calculations that previously took months with a cluster can now be performed in several hours with a laptop. This, according to the authors, opens a new era of interaction between traditional calculus and the most modern computational tools.

Likewise, although the equations have been studied in a confined case – inside a cylinder, as proposed by Luo and Hou –, the conclusions obtained can serve to better analyze the general case – even in situations modeled by other equations. In particular, it is possible that they can also be applied to study the Navier-Stokes problem – the millennium problem – which requires that the fluid not be confined, among other technical differences. Therefore, we cannot yet affirm that this type of machine learning techniques will find the coveted singularity of the Navier-Stokes equations, but the latest results in this direction are very promising and, perhaps, we just need to find the candidate solution. suitable to see the famous problem solved.

Javier Gómez is a professor at Brown University (USA).

Café y Teoremas is a section dedicated to mathematics and the environment in which it is created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share points of encounter between mathematics and other social and cultural expressions and remember those who marked its development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”

Ágata Timón G Longoria is coordinator of the Mathematical Culture Unit of the Institute of Mathematical Sciences (ICMAT).

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