OpenAI Says AI Model Solved 200-Year-Old Navier-Stokes Problem

OpenAI says an internal artificial intelligence model has solved the Navier-Stokes problem, one of the seven Millennium Prize Problems regarded as among the most challenging unsolved questions in mathematics.

The company made the claim in a statement released Tuesday, saying its unreleased model completed a solution to the 200-year-old mathematical problem involving the behavior and motion of fluids.

According to OpenAI, the system took about 88 hours to work through the problem and used as many as 10,000 AI agents operating partly concurrently.

“This milestone represents substantial work by mathematicians and AI researchers. However, this is not a culmination, but rather a snapshot in time, of progress on AI development,” the company said.

The announcement has, however, drawn attention within the mathematics community over questions surrounding the development and verification of the claimed solution.

The night before OpenAI's announcement, New York University mathematics professor Tristan Buckmaster released a statement saying he and Anthropic researcher Levent Alpöge had also been working on the Navier-Stokes problem.

Buckmaster said their work had been ongoing since the previous year and involved several large language models, including OpenAI's Codex. He said the researchers contacted OpenAI as they prepared to make their own results public and subsequently learned that OpenAI researchers were preparing to announce a similar development.

Buckmaster stressed that he was not accusing OpenAI of using his team's work.

“I have not seen OpenAI’s proof. I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything,” he said.

OpenAI said its work on the problem began on September 1 after researchers heard what the company described as a rumour that another Millennium Prize Problem had been solved.

The company said its AI agents had access to tools including a cached version of the internet and the ability to run code. However, OpenAI said the agents did not see Buckmaster and Alpöge's work before it was publicly released and that no specific user data was accessed to solve the problem.

OpenAI said it later offered Buckmaster and Alpöge the opportunity for a concurrent release of their results, as well as access to the prompts used by its system and the opportunity to examine the proof.

The company also released a Lean formal proof alongside its announcement.

However, OpenAI acknowledged that it could not completely rule out the possibility that de-identified data from users of its products may have contributed to improvements in its models.

The development has also prompted broader discussion about the role of artificial intelligence in mathematical research.

Terence Tao, a mathematics professor at the University of California, Los Angeles, said the use of AI in advanced mathematical research could change how mathematicians' approach difficult problems. He also expressed concern that excessive reliance on AI could reduce the value of the process of mathematical discovery.

The Navier-Stokes problem concerns equations used to describe the motion of fluids and asks, in part, whether solutions to those equations remain smooth under relevant conditions.

The problem is one of seven Millennium Prize Problems selected by the Clay Mathematics Institute in 2000. The institute offered a $1 million prize for a correct solution to each problem.

So far, only one of the seven problems—the Poincaré Conjecture—has been officially recognised as solved.

OpenAI said it does not intend to claim the $1 million Millennium Prize for its reported Navier-Stokes result.

The company's announcement therefore represents a significant claim about the capabilities of AI in advanced mathematics, while the mathematical community continues to examine and assess the reported solution.