Mathematicians have long grappled with the Navier–Stokes equations, the core mathematical description of fluid motion in physics and engineering. These equations model things such as water flowing through a pipe or air moving over an airplane wing, but one of their most famous open questions is whether solutions can “blow up,” that is, become infinite or nonsensical under some conditions. A proof either way carries a million-dollar prize from the Clay Mathematics Institute.
In a Quanta Magazine article, researchers reported progress not on the full Navier–Stokes problem but on simpler versions of fluid equations, where they have now identified previously unseen patterns where solutions might break down. The innovation comes from physics-informed neural networks (PINNs), a class of AI models trained not on image or language data but directly on the rules embedded in the equations themselves. These networks adjust themselves until they satisfy the equation, revealing candidate singularities by finding functions that could lead to infinite behavior.
Traditionally, mathematicians used numerical simulations that evolve fluid configurations over time. But those methods struggle to detect “unstable singularities,” subtle breakdowns that only appear under extraordinarily precise initial conditions and are wiped out by tiny variations. By contrast, PINNs do not simulate time evolution; they target solutions directly. That makes them much better at identifying unstable candidates because the network can focus on solutions themselves rather than the path to get there.
The team rediscovered known singularities and found many new ones in a range of simplified fluid models, including unstable cases in more than one dimension, a first of its kind in those contexts. While none of the new candidates has been proved rigorously yet, the precision of the network outputs makes formal proofs more attainable than before.
This work does not yet resolve the Navier–Stokes challenge, but it represents a substantial shift: machine learning is proving itself a powerful tool for exploring deep mathematical problems that resisted conventional analysis for decades.