OpenAI said today that it has found an AI-generated solution to one of the biggest problems in mathematics—a 200-year-old equation that describes the natural behavior of fluids like water and air.
The announcement appears to demonstrate the stunning power for AI to advance mathematics. But it has been marred by claims from another mathematician, Tristan Buckmaster, who says that OpenAI rushed ahead to solve the problem after learning of his and another mathematician’s progress on the problem, and also then tried to influence who got credit for the work.
The proof concerns the Navier-Stokes equation, one of the unsolved problems in the Clay Millennium prizes, which are each worth $1 million.
Sebastien Bubeck, a mathematician and AI researcher at OpenAI, said in a press briefing that the company began training a new AI model with advanced mathematical capabilities on August 28.
After reading rumors that Anthropic was making progress towards solving Navier-Stokes, Bubeck said the company decided to dedicate more resources to tackling the problem. The company had over a thousand agents tackle the problem over more than 50 hours, before discovering that it had come up with a solution.
“I thought there must be a mistake somewhere,” Bubeck said. “And on Sunday morning we had the final solution, Lean-formalized, and everything.” (Lean is a programming language that can be used to formalize mathematical proofs.)
OpenAI noted that solving this problem required using considerably more compute than it had previously spent on solving mathematical problems. The amount of compute required cost “in the millions of dollars,” Mark Chen, head of research at OpenAI, said.
On Monday, Buckmaster, a mathematician at NYU, and Levent Alpöge, a researcher at Anthropic, posted documents claiming key advances in an area relevant to the Navier-Stokes problem. The pair says that they used several AI models, including Claude and Codex, to complete their work.
Buckmaster also posted a statement claiming that, last week, he learned that OpenAI had become aware of his and Alpöge’s work, and had started putting significant resources towards the problem. Buckmaster claims that he asked OpenAI leaders about whether the company had accessed the pair’s Codex logs. He says he was told the model “didn’t look up user data,” but claims the company didn’t respond to questions about training. He then says that OpenAI offered several “proposals,” including one in which Buckmaster could publish a paper announcing the Navier-Stokes problem had been solved by an internal OpenAI model, but without Alpöge’s name included.
Buckmaster, Alpöge, and Anthropic did not immediately respond to WIRED’s request for comment.
In the briefing, Bubeck and other OpenAI executives denied that the company had ever inspected the pair’s Codex prompts in order to inform their work. “We, whether it’s the researchers or the agents, did not see any of their work until it was released publicly last night,” he said.
“I want to be extremely clear that we recognize the priority of Levent Alpöge and Tristan Buckmaster’s work on unforced Euler, and we have nothing but congratulations to them on this monumental achievement that they have made. To be clear, we did not use their prompt or proof to prompt our models or direct our agents,” said Bubeck.
OpenAI said today it wanted to recognize Alpöge and Buckmaster’s prior work. However, Ven Chandrasekaran, a mathematician at the company, also emphasized that their solution was significantly different in nature to the one produced by OpenAI’s model.
The spat over who gets credit for solving one of the biggest problems in math may rumble on for a while. As AI takes on more of the work involved with finding proofs, such fights could perhaps become more common.
This is a developing story. Please check back for updates.