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

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

If the result holds up to further scrutiny, it is, by a significant margin, the most important mathematical proof to have been arrived at by an artificial-intelligence model to date, possibly marking a fundamental turning point in how mathematicians tackle difficult problems.

This particular difficult problem deals with differential equations, which express relationships between changing quantities. They are arguably the single most important mathematical tool for explaining the world around us. As a rule, they are easy to write down and hard to solve.

The Navier-Stokes equations are differential equations that use Newton’s second law of motion to describe how fluids, from ocean currents to air flows, behave. They were first written down in the mid-19th century, and have been central to the study of fluid mechanics ever since. But one basic question about the equations has persisted: Are their solutions always well-behaved? Or can their solutions evolve over time so that some infinitesimally small part of the fluid begins to flow infinitely quickly, creating a so-called singularity?