Discovering new solutions to century-old problems in fluid dynamics

Fluid dynamics has a dirty secret: the equations we use to model swirling water, air, and hurricanes sometimes break down completely. Under very specific, contrived conditions, values like pressure or velocity shoot to infinity — a phenomenon mathematicians call a singularity, or a blow up. These moments are physically impossible, but they're mathematically important, because they expose the edges of what our equations can actually describe.

DeepMind's newest paper, done with researchers from Brown, NYU, and Stanford, digs into a particularly nasty subset of these: unstable singularities. Unlike stable ones, which pop up under a wide range of conditions, unstable singularities need everything lined up almost perfectly. That precision requirement is exactly why they're so hard to find by hand, and why nobody had systematically catalogued new families of them until now. It matters because many mathematicians suspect that for the boundary-free 3D Euler and Navier-Stokes equations — the ones tied to one of the six unsolved Millennium Prize Problems — no stable singularities exist at all. If that's true, the unstable ones are the whole ballgame.

The team's tool of choice was a variant of physics-informed neural networks, or PINNs. Normally these networks get trained on data. Here, there's no data — just the physics itself. The network's output gets checked directly against the governing equations, and it learns by shrinking a