Terence Tao calls AI-assisted fluid dynamics proofs ‘a remarkable achievement’

NYU mathematician Tristan Buckmaster and Anthropic researcher Levent Alpöge on Sunday published AI-assisted, Lean-verified proofs showing that several fluid equations can develop finite-time blow-ups — points where physical quantities spiral to infinity — under smooth external forcing. Fields Medalist Terence Tao called the work "a remarkable achievement" and said he sees no fundamental obstacle to extending the methods toward the Navier-Stokes equations, one of mathematics' seven unsolved Millennium Prize Problems.officechai+1

What the Proofs Show

The results cover the 3D incompressible Euler, Boussinesq, and incompressible porous media equations. Prior work by mathematicians Diego Córdoba and Luis Martínez-Zoroa had demonstrated blow-up was possible only with rougher, less physically realistic forcing terms. Buckmaster and Alpöge say they pushed the forcing all the way to smooth — a harder and more meaningful result — with heavy assistance from AI models and formal verification in Lean.terrytao.wordpress+2

Tao, writing on his blog, explained the underlying strategy: the construction iteratively layers small, high-frequency corrections onto a larger background flow, with the background exponentially amplifying those corrections until they become strong enough to trigger the next round of amplification. Repeated fast enough, this produces a genuine finite-time blow-up. He noted the Boussinesq paper alone runs 76 pages despite efforts to trim it.officechai+1

A Path Toward Navier-Stokes?

Tao said he sees nothing in the current method that would fundamentally block an extension to the Navier-Stokes equations and floated the possibility that the forcing term could eventually be removed entirely, which would move the result closer to the actual Millennium Prize formulation. But he cautioned that substantial technical difficulties remain between the current results and a full resolution. He said he would not be surprised if someone eventually pushed through those difficulties with enough compute and AI assistance, but added that the grind does not particularly interest him personally — he is more excited about digesting the new proof techniques for their mathematical content.terrytao.wordpress+1

Controversy Surrounding the Release

The publication was accompanied by allegations from Buckmaster that OpenAI attempted to pressure him during the release process after rumors circulated linking the work to a possible Navier-Stokes breakthrough. OpenAI's Sebastien Bubeck denied wrongdoing in a brief public response, promising a fuller account. Separately, a research group at Anima AI disclosed it had been independently working on related blow-up results for unforced incompressible Euler equations using a different approach based on physics-informed neural networks.anima-ai+2

Tao noted that he spoke to Buckmaster directly by phone to have the ideas explained — a detail he contrasted with AI-mediated communication, calling the human conversation "a refreshing change of pace".officechai