🧮 AI agents solved one of mathematics' hardest problems in 88 hours

• OpenAI says around 10,000 AI agents found a solution to the Navier-Stokes problem in 88 hours, a problem mathematicians have worked on for decades.
• The problem is one of six Millennium Problems listed by the Clay Mathematics Institute, each carrying a prize of $1 million.
• If the result is confirmed, it is the first time AI has solved a major open problem in mathematics.

Equations from the 1800s

Nearly 200 years ago, Claude-Louis Navier and George Gabriel Stokes set out to describe the motion of fluids mathematically. The equations they wrote down are still used routinely, by everyone from aircraft designers to climatologists. They describe the physics of flowing fluids such as water, and in some cases air. Blood in a heart or magma in Earth's mantle can be described by noting the fluid's velocity and pressure at every point in the volume.

The equations were not guaranteed to work under all possible conditions. For decades, mathematicians have tried to prove either that the equations always give a physically realistic answer, or that there are situations in which they break down.

A vortex with infinite speed

OpenAI's agents found a solution to the equations that, in a finite amount of time, evolves into a whorl in which the fluid moves at infinite speed. Nothing can move infinitely fast, not even light. They therefore showed by example that there are cases in which the equations produce an answer with no physical meaning.

That does not mean the equations are wrong, but that they are incomplete. They assume the fluid is a smooth, continuous medium, while a real fluid consists of discrete atoms and molecules. The result shows that it is possible to construct a scenario in which the equations wander out of their own range of applicability, and that additional physics then has to kick in.

Started with a simpler version

Rather than attack the difficult problem head on, the agents started with the unforced Euler equations, a simpler cousin that assumes the fluid has no viscosity. After 50 hours of work by 1,000 agents, they had a solution. OpenAI then put 10,000 agents on the full Navier-Stokes problem. By the morning of 5 September there was a solution, and by the next day it had been formalized in Lean, a programming language that can automatically verify mathematical proofs.

The work was carried out by an internal model that OpenAI describes as significantly more capable than GPT-6 Astra. Training of the new model began on 28 August. Chief Scientist Jakub Pachocki said that a year ago OpenAI was still looking at whether its models could compete with high schoolers in mathematics olympiads. He described the work as an evaluation of a system for general purpose intelligence, not as a long-running effort aimed at the Millennium Prize problems.

Millions of dollars in computing power

The effort was far larger than OpenAI's previous math experiments. The Astra system cost about $2,000 for ten results on other mathematical problems. The Navier-Stokes effort cost roughly a thousand times more, in the ballpark of millions of dollars, according to Chief Research Officer Mark Chen.

Karthik Duraisamy at the University of Michigan estimates that the same work would cost about $6 million for anyone buying OpenAI's tools at retail rate. Since OpenAI already has the model and the resources, its own cost comes to about $1 million in inference compute, the same amount as the prize money. OpenAI researcher Sébastien Bubeck said the company can now spend millions of dollars on problems such as developing new materials and finding cures to diseases.

Parallel work on the same problem

Mathematician Tristan Buckmaster of New York University and researcher Levent Alpöge have spent the past year using a variety of AI tools to tackle the Navier-Stokes problem. Last month their AI found a solution to the Euler equations and verified it in Lean. Since then the pair have worked through the proof and translated it into a form mathematicians can read and assess. They released their work on 7 September.

The release has touched off a dispute over who deserves credit for the solution. Buckmaster has questioned OpenAI's account of what the company's system had access to, after he says he was offered sole authorship if Alpöge's name was removed. OpenAI denies that any employee or AI agent saw the pair's work before it was published, while also stating that it cannot rule out that de-identified data derived from the pair's use of its products helped improve its models.

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