Caltech-affiliated team claims stable singularity in three-dimensional Euler equations

Anima AI researchers affiliated with Caltech say they have found a stable singularity in the three-dimensional Euler equations, one of the most stubborn open problems in fluid mechanics and applied mathematics. The equations describe ideal, inviscid flow; whether they can develop singularities in finite time has been debated for decades. If the claim survives peer review, it would represent a significant step toward proving finite-time blowup exists.

The approach combines a physics-informed neural network (PINN) with a stability argument. The PINN generates an approximate solution, and the stability analysis around that solution completes the proof. PINNs have previously failed to uncover singularities in this problem, typically collapsing to trivial solutions. The team developed a method to steer the PINN toward non-trivial regions by layering constraints — a technique they believe could help PINNs tackle other hard optimization problems of this class.

Beyond constructing the approximate profile, the researchers analyzed its transport field in detail. They report that the field shows promising local outflow properties, a key ingredient for proving linear damping, which in turn underpins the overall stability. The link between the PINN-derived profile and the transport-field analysis is what they say closes the proof loop.

The researchers argue that physics-informed, physics-focused AI is essential for many domains dealing with physical systems, and that LLMs lack this physical grounding. They have worked on the problem for most of the year. They note that Tristan recently announced results on the forced Euler equations, whereas their work addresses the unforced case using a PINN formulation.