OpenAI Paper Proposes Navier-Stokes Blow-Up, PDE Experts React
The striking implication is that an AI lab produced mathematics that leading PDE researchers describe as possibly beyond what a human would have devised, with the mechanism lying outside prior literature. Researchers in mathematical fluid dynamics and anyone tracking AI's role in formal proof work should watch for the expert digests Armstrong expects, then compare the construction against Elgindi's blow-up results and the Albritton–Brué–Colombo framework.
An OpenAI research paper proposes a construction for a blow-up (non-uniqueness) solution to the 3D Euler/Navier-Stokes equations, drawing intense attention from fluid dynamics experts. Mathematician Scott Armstrong responded that the construction relates to convex integration, using a fluid instability that draws energy from background shear so the flow itself generates oscillations. He noted the technique has no precedent, though related ideas appear in Leibovich-Stewartson 1983 and the Albritton–Brué–Colombo non-uniqueness construction. Armstrong says several PDE expert groups now more or less understand the paper and expect detailed digests soon.
“On a technical level, the paper is unbelievable, maybe beyond what a human could have come up with but not beyond human understanding” Scott Armstrong @scottnarmstrong Since I'm now blocked by bluey from replying to this thread, I answer here by quote tweet the openAI construction has more to do with convex integration than anything else. In convex integration, the oscillations are typically built by hand and the iteration procedure creates wild weak solutions. The non-uniqueness construction of Albritton–Brué–Colombo used an unstable vortex to split solutions apart. The Córdoba–Martínez-Zoroa singularities amplify vorticity across scales by strain. In the openAI paper, a fluid instability is clever made so that the flow itself generates the oscillations: they grow by drawing energy from the background shear. The Reynolds stress then becomes the force that creates the collapsing smooth vortex. Using a physical instability to create the stress in this way has no precedent, and it is what produces a smooth force rather than a merely rough one. Some on these ideas are in the literature (e.g. Leibovich-Stewartson 1983) but the way they are combined is definitely insane and definitely nowhere in the literature previously. I also don't know of anyone suggesting that convex integration would play a role in the creation of a blow-up solution like this. On a technical level, the paper is unbelievable, maybe beyond what a human could have come up with but not beyond human understanding: there are several different groups of pde experts working on understanding it who say that they more or less understand now. I expect digestions soon td;lr: anyone saying that this OAI result is unsurprising is bullshitting sorry! 🔗 View Quoted Tweet 💬 21 🔄 8 ❤️ 89 👀 10566 📊 21 ⚡
- Nathan Lambert11:19原文
