OpenAI has claimed that an advanced internal AI system has solved the Navier-Stokes existence and smoothness problem, one of mathematics' seven Millennium Prize Problems. The company says its proof shows that a smooth fluid can develop a singularity, where speed becomes unbounded in finite time. The claim has not yet been independently verified by the wider mathematical community.
According to The New York Times, the announcement is the latest sign that AI is transforming the highest levels of mathematics, a field long regarded as a hallmark of human intellectual achievement. While some mathematicians have welcomed the development, others have expressed concerns about its implications.
OpenAI stated that an unreleased model required 88 hours to solve the "Navier-Stokes existence and smoothness problem." The equations describe the movement of fluids and are often used to predict the weather. The problem asks whether these equations completely break down in certain situations, and OpenAI's proof claimed to define such a case.
Across all attempted problems, the agents sent 4.9 million messages and used about 300 billion output tokens. In the process of resolving the Navier-Stokes problem, the agents sent 2.7 million messages and used approximately 130 billion output tokens.
"This is a spectacular culmination of the arc we have seen over the past twelve months," The New York Times quotes OpenAI researcher Sebastien Bubeck.
The Clay Mathematics Institute selected the Navier-Stokes problem in 2000 as one of seven Millennium Problems, offering a million dollars for each solution. Before this announcement, mathematicians had solved only one of the seven.
"These questions are lighthouses," the news report quoted Terence Tao, a mathematics professor at the University of California, Los Angeles. "They are great focus points that attract the efforts of human scientists."
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Tao warned that such technologies risked weakening human understanding of mathematics.
"The effort needed to solve problems is often very instructive. It teaches you something. It's like going to the gym and having a goal to lift a weight a hundred times," Tao explained. "Now, AI can solve questions without really getting any value out of them. It's like having machines that can lift weights for you at the gym."
OpenAI stated that it solved the problem by deploying as many as 10,000 "A.I. agents" working together, with researchers relaying key ideas between teams.
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"Our role was like a bumble bee cross-pollinating across different groups and delivering different bits of information," the news report quoted OpenAI researcher Dan Roberts.
The news report mentioned that OpenAI research scientist Noam Brown called the computational operation a "very expensive process," though he noted that operating costs generally drop as efficiency improves.
The company acknowledged that it focused resources on the Navier-Stokes problem after learning that other mathematicians examined similar research, though it stated that it "did not see any of their work through any means."
The systems rely on reinforcement learning to generate formal mathematical proofs in a programming language called Lean. On Tuesday, OpenAI released a paper outlining its Lean proof to allow outside verification. Tao compared the automated approach to a guide who finds a single route to a waterfall.
"This has some value," Tao said. "But once someone shows a specific path to the waterfall, people just take that path. They don't spend as much time looking for other paths."
The American Math Society welcomed the announcement from OpenAI and said the process that led to solving the problem will bear fruit for a long time to come.
"The news today of progress on resolving the Navier-Stokes problem, one of mathematics' great longstanding challenges concerning the equations that govern the flow of fluids, represents a milestone advance in human knowledge. This story began with Navier, Stokes, Leray, and Ladyzhenskaya and has culminated in the recent breakthroughs of Córdoba and Martínez-Zoroa, then - assisted by new technologies - Alpöge and Buckmaster, with the final steps taken by OpenAI mathematicians. The purpose of mathematics is human understanding, and this achievement, and the process that led to it, will bear fruit for a long time to come," the society said.
(With inputs from Reuters)
