AI helped find fragile new ways fluid equations break down: precise numbers, not proofs
Researchers at Google DeepMind, Brown, NYU and Stanford used AI to find them. They give mathematicians precise leads, not proofs.
Equations for fluid flow can sometimes blow up: the solution develops a singularity, a point where it stops making sense. A paper posted on 17 September 2025 (arXiv 2509.14185) reports the first systematic discovery of new families of unstable singularities. These are breakdowns that happen only under extremely precise conditions.
The 22-author team, from Google DeepMind, Brown, NYU and Stanford, found several new solutions of this kind for the incompressible porous media equation and for the 3D Euler equation with a boundary. DeepMind’s blog post adds the Boussinesq equation. They used physics-informed neural networks (networks trained to satisfy the equations) with a high-precision optimiser. Some solutions come close to double-precision machine accuracy, the limit of standard computer arithmetic, which the authors say meets the bar for computer-assisted proofs. Co-authors include Tristan Buckmaster and Javier Gómez-Serrano.
Why unstable ones matter: DeepMind notes that mathematicians expect no stable singularities for the 3D Euler and Navier–Stokes equations without boundaries. So any breakdown there would have to be an unstable one. This was groundwork for the rigorous blow-up proofs that followed in 2026.
What it means
AI has become a practical tool for finding hard-to-spot solutions in maths and physics, which people can then try to prove. These results are not proofs, and they do not settle the famous Navier–Stokes question. They give mathematicians precise candidates to work from.
Caveats
- The results are numerical; the paper does not claim a rigorous proof.
- The 2026 forced blow-up proofs used a different, forcing-based approach rather than this method directly.
- We found no 2026 DeepMind computer-assisted proof of an unstable singularity.