AI Has Solved One of Math’s $1 Million Millennium Prize Problems

On September 8, 2022, OpenAI reported that their 10,000 autonomous AI agents had discovered a solution to one of mathematics' most difficult unsolved problems, the Navier-Stokes equations. This discovery earned the team a $1 million prize from the Clay Mathematics Institute. The Navier-Stokes equations are a set of differential equations that describe the behavior of fluids such as air and water.

They have been a central focus of fluid mechanics since the mid-19th century. Despite their importance, a fundamental question remained unanswered: do the solutions of these equations always remain well-behaved, or can they develop singularities where the fluid flow becomes infinitely fast? Researchers at OpenAI used a special AI model not available to the public to find a "singularity" in the equations.

This breakthrough was later verified using the Lean programming language, increasing the confidence in its correctness. If this result holds up, it would be a groundbreaking achievement, marking a major turning point in the way mathematicians approach complex problems. The Navier-Stokes equations are based on the assumption that fluids can be divided infinitely, but in reality, fluids are made up of molecules and atoms.

The discovery of a singularity, even in an idealized sense, highlights the surprising complexity of fluid behavior and the profound consequences of even simple laws like Newton's second law of motion. The findings also open up new questions about turbulence and the counterintuitive nature of fluid dynamics. The OpenAI researchers relied on the work of Diego Córdoba and Luis Martínez-Zoroa, who developed a new strategy to tackle the problem, earning Martínez-Zoroa a potential Fields Medal.