A stable singularity in 3D Euler equations, found with a neural net

Prof. Anima Anandkumar@AnimaAnandkumar

We have found stable singularity on 3D Euler!
t.co/40miuTjMvl

Our starting point is a physics-informed neural network (PINN) to come up with an approximate answer, and then to argue stability around that to complete the proof.

The challenge so far has been that PINNs have not been successful in discovering singularities on the current problem. A common failure mode is PINNs converging to a trivial solution. We take special care to nudge our PINN to interesting regions through a combination of constraints, and we believe this is a novel way of making PINNs work for such hard optimization problems.

We also carry out a detailed study of the transport field of our approximate profile, and show it has promising properties of local outgoing flow, essential for proving linear damping, which is an important ingredient of overall stability.

We believe that such physics-informed and physics-centric AI are critical ingredients across many areas of research involving physical systems, and LLMs lack such physical grounding.

We have been working on this problem for much of this year, and we just saw the announcement by Tristan on Euler with forcing. In contrast, we consider without forcing and use a PINN formulation. @Caltech