AI finds new singularities in fluids

DeepMind and several groups of mathematicians have identified new families of unstable singularities in equations that describe fluid motion. The finding does not solve the Navier-Stokes problem, but it offers a new way to explore one of mathematics' major open challenges.

Fluid dynamics equations try to describe phenomena such as a hurricane's vortex, airflow around an aircraft, or the movement of water. In certain mathematical scenarios, their variables can grow without limit: velocity or pressure, for example, could tend toward infinity.

These situations are known as singularities or blow ups. They do not mean that a real fluid will reach infinite values. They are extreme cases that make it possible to study how far the equations work and what limits they have when representing the physical world.

A problem that has remained open for decades

A singularity is stable when it survives small changes in the initial conditions. If it requires an extremely precise configuration, it is considered unstable.

Researchers believe this second type could be important for understanding some fundamental equations in fluid dynamics. In particular, it has not yet been proven whether three-dimensional Navier-Stokes equations can generate a singularity. Solving that question is one of the six Millennium Problems, each carrying a prize of one million dollars.

The new work, carried out with mathematicians and geophysicists from institutions including Brown University, New York University and Stanford University, presents the first systematic AI search for new families of unstable singularities in three different fluid equations.

In two of them, the equations for incompressible porous media, known as IPM, and the Boussinesq equations, a pattern also appeared. As instability increases, the speed at which the blow up occurs, represented by the Greek letter λ, appears to follow a line. This suggests that more solutions may exist that have not yet been found.

How AI helps

The team used physics-informed neural networks, known as PINNs. Unlike an AI trained mainly on large collections of data, these networks receive the equations themselves as a guide.

The model proposes a solution and continuously compares it with what the laws of physics should require. It then adjusts its parameters to reduce the error, called the residual, until the answer fits the equations.

In this case, the researchers did not use PINNs only to obtain an approximation. They incorporated mathematical knowledge about singularities and used second-order optimization methods, a technique that allows training to be refined with greater precision.

The result reached an accuracy close to the limit of the available machines. For reference, the largest corrected errors are equivalent to calculating the Earth's diameter with a difference of only a few centimeters.

What changes in practice

You will not notice an immediate change in weather forecasting or aircraft design. The advance lies elsewhere: it makes it easier to explore mathematical solutions that traditional methods struggle to find.

That could be useful for:

• Studying the limits of the Navier-Stokes equations and other fluid models.
• Designing computer-assisted proofs with a verifiable level of precision.
• Investigating problems in mathematics, physics and engineering that require many numerical checks.

The important difference is that AI is not presented here as a black box that gives you an answer and expects you to accept it. The model is required to respect the equations, and its results can be analyzed with mathematical tools.

The next step will be to verify the families that were found more rigorously and determine whether the observed pattern holds in other solutions. The work points to a new collaboration between mathematical intuition, simulation and AI, but it remains to be shown what definitive consequences these singularities have for the most difficult equations in physics.