Some Turbulent News
This post is about the recent excitement generated by OpenAI's purported solution of the Navier-Stokes problem, which carries a million dollar Clay Mathematics Institute Millennium prize. I thought it would be especially appropriate to write about it as I am teaching a graduate course on solving partial differential equations this semester.
Differential Equations - Linear and Nonlinear
The laws of physics are quantified using differential equations (DEs). DEs use rates of change (derivatives) to connect how physical properties change across time and space. Thus, they provide the natural language for describing physical reality. Popular examples are Newton's laws and Maxwell's and Schrodinger's equations.
If the DE involves one independent variable (e.g. space or time) then it is said to be ordinary (ODE); if more than one variable is involved, then it is said to be partial (PDE). DEs are also classified as being either linear or nonlinear in their dependent variable (the sought after solution) [1]. As a rule of thumb, linear DEs are easier to solve than the nonlinear ones.
The Navier-Stokes equations
The Navier-Stokes equations (NSEs) are a set of nonlinear PDEs that govern the motion of viscous fluids (as in the wake of the boat in the image above). The NSEs are very important to both science and technology as they predict ocean currents, weather patterns, aerodynamic drag on aircraft, blood flow, pollution dynamics, industrial fluid flow in pipes, power stations, and vehicles, and so on.
The NSEs are an approximation: they ignore the molecular reality of fluids (liquids and gases are ultimately made of discrete atoms) and treat them as continuous bodies. Since the solutions to the NSEs are important, for about a century, mathematicians have famously tried to determine whether this 'continuum' approximation (and hence the NSEs) breaks down, and the solutions display singularities.
The answer to this problem is tied to one of the most fascinating and famously unsolved phenomena in physics: turbulence - fluid motion characterized by chaotic, rapid changes in pressure and velocity, as opposed to smooth (laminar) flow. This problem is intractable using pen-and-paper techniques (such as the ones I am teaching this semester) or even conventional computer simulations.
Enter AI
AI has fundamentally transformed the analysis of NSE, in two major ways. In applied physics, neural networks have replaced months-long complex fluid dynamics mesh simulations with near-instantaneous neural inferences. In formal academic mathematics, the multi-agent computing paradigm has proved that AI can navigate abstract mathematical landscapes [2].
About 3 days ago, OpenAI announced that an unreleased multi-agent system had successfully resolved the smoothness problem for NSEs. Operating about 10,000 autonomous AI agents over 88 hours, the system discovered a disproof of global smoothness (establishing statements "C" and "D" of the official Clay Mathematics Institute formulation).
AI proved analytically that an initially smooth fluid driven by a smooth force can collapse into a finite-time blowup, and hence is not always smooth. Analyzing the NSEs, AI described a spaghetti-like vortex that spins faster and narrows progressively, causing the local fluid velocity to reach infinity while keeping the overall energy finite—abiding by physical conservation laws.
Following the 166-page analytical proof (I love their singe-sentence abstract), the system used GPT-6 Astra to write and verify the entire proof in Lean (a theorem prover; I enjoyed learning that one of their automation tactics is called Grind - reminds me of 'grinding' through a proof), in about 17 hours.
Summary
The result has not yet been checked by human mathematicians. OpenAI has said it has no intention of collecting the $1M Clay prize (the computation itself cost a few million); I guess it is playing for far larger stakes. Stay tuned for more!
[1] Technically speaking: A differential equation is linear if the dependent variable and all of its derivatives appear only to the first power, are not multiplied together, and do not appear inside transcendental functions.
