OpenAI's AI Agents Solve a $1 Million Millennium Prize Math Problem

On Tuesday, September 8, mathematicians at OpenAI announced that 10,000 autonomous AI agents, working under their direction on an advanced model not available to the public, had found a singularity in the Navier-Stokes equations in three dimensions. The result resolves one of the six remaining Millennium Prize Problems, posed in 2000 by the Clay Mathematics Institute, each carrying a $1 million prize. The proof has been formally checked in the programming language Lean, giving mathematicians confidence that it is correct.

The Navier-Stokes equations are differential equations, based on Newton's second law of motion, that describe how fluids such as ocean currents and air flows behave. First written down in the mid-19th century, they have been central to fluid mechanics ever since. A basic open question was whether their solutions are always well-behaved, or whether an infinitesimally small part of a fluid could begin flowing infinitely quickly, creating a singularity. OpenAI's solution takes the form of a vortex.

OpenAI's announcement came 12 hours after a separate one from Tristan Buckmaster of New York University, who said he and Levent Alpöge of Anthropic had resolved several related problems with help from various AI models, including OpenAI's. Both teams relied on a strategy developed by Diego Córdoba of the Institute for Mathematical Sciences in Madrid and Luis Martínez-Zoroa of CUNEF University, which departed radically from the methods most mathematicians had used.

Charles Fefferman of Princeton University, who wrote the Clay Institute's official description of the Navier-Stokes problem, said he was thrilled that the problem was solved, calling Córdoba and Martínez-Zoroa the heroes of the story. Buckmaster said he believes Martínez-Zoroa deserves a Fields Medal.

In 2013, Thomas Hou of the California Institute of Technology and Guo Luo of the Hang Seng University of Hong Kong showed that the related, frictionless Euler equations can "blow up" inside a cylinder whose top and bottom halves spin in opposite directions.