Self-similar study of stable singularity in unforced Euler equations on R^3

Summary

[This is a guest post by Anima Anandkumar . This blog post was initially written in a different file format and converted using AI. — T.] The last few days have been a flurry of activity. After months of sleepless nights and diving deep into different mathematical and computational tools to tackle the open problem of singularity in fluid dynamics, we were finally ready to send our work on Euler equations (without forcing) to Tom Hou and Terry Tao for feedback. Almost immediately I heard back from Terry encouraging us to release our work publicly, given that the NYU team had just released their results on forced Euler and OpenAI was rumored to release the next day. We scrambled and put together a blog post and attached our paper with code, which went live about 30 minutes after, in the evening of September 7, the day before the OpenAI’s announcement. A lot of mainstream media religiously followed their press release that fails to acknowledge our work even after we informed them. On the bright side, many researchers have reached out to me. I want to thank Terry for highlighting our work, as well as for giving me the opportunity to write this blog post. The starting point for us is different from the route that the NYU team and OpenAI pursued. Their initial solution has a specific construction with a multi-scale structure with discrete jumps in scales. On the other hand, we consider a self-similar ansatz, and so far, no one has been able to construct a self-similar singular profile for the Euler equations in without forcing. Modified cases such as the Euler equation with a boundary have been successfully solved before ( Chen and Hou ), where the boundary helps confine the singularity, but such techniques fail in free space. We wanted to see if AI could come up with a self-similar profile for Euler. LLMs are not suited for generating such singularity candidates without analytical closed forms. Physics informed neural networks (PINN), on the other hand, can be optimized to find such solutions. Although there have been prior attempts to use PINNs for this case, they were not successful. There are optimization and precision challenges: with standard ansatz and optimizers, a common failure mode is getting the trivial solution. The other challenge is getting precision to be high enough, so we can then attempt to convert the numerical solution into an analytical proo…