Open AI claims to have constructed an analytical solution to the Navier-Stokes equations which starting from zero under smooth forcing ceases to exist in finite time:
- We’re sharing a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems. This proof, produced by an internal OpenAI system, shows that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time. We’re sharing both a writeup of the proof and a formalization in Lean.
Charles Fefferman, who formulated the problem in precise mathematical terms, is along with other leading mathematicians such as Terence Tao, happy that the understanding of fluid motion has now taken a big leap forward by mathematical analysis.
There is one problem in this happy moment, which I have complained about over the years: Fefferman's formulation misses the essence of the physics of fluid motion, namely turbulence. The Clay problem is sold as concerned with basic aspects of fluid motion, but does not address the most fundamental problem of all of turbulence. Fefferman's formulation directs the interest away from physics, and the unhappy result is that solution now presented by AI covering 167 pages cannot be read to learn anything, simply a mess of formulas and theorems.
This is certainly a memento for analytical mathematics: AI can now produce analytical proofs of an endless number of mathematical problems without real meaning, proofs which cannot be understood by mathematicians in detail only verified formally by Lean. What will be the result?
Computational mathematics offers a solution to the fundamental problem of turbulence, thus a different solution to a different formulation. See tags to this post starting with this post from 2013.
Recall that slightly viscous flow is unstable from shear and stretch and so develops into non-smooth turbulent flow which however does not break down like the Clay solution. So the solution of physical interest is non-smooth and non-singular, which is not captured in Fefferman's dichotomy of smooth or singular.
Turbulence is an extreme form of the design of a complex world with a variety of phenomena on different scales: Develop growth from instability + curb growth by to allow continued existence, not captured by Fefferman's formulation.
PS When I 20 years ago complained to Fefferman that his formulation lacked true interest from physics point of view, he returned that it was enough that the problem was interesting to him.
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