Discovering new solutions to century-old problems in fluid dynamics

Source: DeepMind Date: 2025-10-24 URL: https://deepmind.google/blog/discovering-new-solutions-to-century-old-problems-in-fluid-dynamics/

Summary

Google DeepMind researchers used Physics-Informed Neural Networks (PINNs) trained to satisfy physical laws — rather than data — to discover new families of unstable singularities in fluid dynamics equations (Navier-Stokes, Euler, Incompressible Porous Media, Boussinesq). The method achieved near-machine-precision accuracy enabling computer-assisted mathematical proofs. A key finding: in two equations, solutions plotted against blow-up speed parameter and instability order aligned along a clear line, suggesting additional undiscovered solution families.

Implications

PINNs at near-machine precision enabling computer-assisted proofs is the methodological advance. Using neural networks as function approximators constrained by physics rather than trained on data is not new. Pushing that approach to machine-precision accuracy — sufficient to generate computer-assisted mathematical proofs — is new. It means the output isn’t “approximately correct” but rigorously verifiable, which is the threshold that mathematicians require for results to count.

Unstable singularities in Navier-Stokes are directly relevant to the Millennium Prize Problem. One of the Clay Mathematics Institute’s seven Millennium Prize Problems is whether smooth solutions to the 3D Navier-Stokes equations always exist or whether singularities (blow-up) can form. Finding new families of unstable singularities doesn’t solve that problem, but it maps the terrain of where blow-up behavior lives — relevant prior work for any future resolution.

The linear alignment pattern across two equations is a discovery that points beyond itself. When you plot solutions in two different fluid equations and they align on the same line, it suggests a shared underlying mathematical structure — and that more solutions exist that haven’t been found yet. That’s the kind of pattern that gives mathematicians a direction to search, which is often more valuable than a single solved problem.

Watch:

• Mathematical follow-up work citing this result — which pure mathematics groups use the newly discovered singularity families as starting points?
• Extension of the PINN + computer-assisted proof method to other unsolved PDE problems (e.g., Yang-Mills, Euler equations in higher dimensions)
• Whether the near-machine-precision PINN methodology gets packaged as a reusable tool for mathematical physics, as AlphaFold’s structure prediction was packaged for biology