OpenAI has claimed that it solved key aspects of a decades-old advanced mathematics problem in just 88 hours using a new artificial intelligence model and about 10,000 autonomous AI agents.
The ChatGPT-maker said its AI agents had resolved key elements of the Navier–Stokes equations, which describe how fluids move and are central to understanding turbulence, a physical phenomenon that remains poorly understood, reports BBC.
Important aspects of the equations have remained without mathematical proof for 90 years.
OpenAI described the achievement as a “milestone” demonstrating the rapid progress of AI tools. The company began training its new internal AI model, a computer programme trained on massive datasets to identify and predict patterns in information, at the end of August.
The model has not been made publicly available because OpenAI considers it “significantly more capable” than its latest public release.
After hearing rumours on 1 September that two Millennium Prize problems had been solved, OpenAI researchers deployed thousands of AI agents trained on the new model to investigate the Navier–Stokes problem.
By 5 September, about 88 hours after assigning 10,000 AI agents to the task, OpenAI said it had reached a solution to the Navier–Stokes existence and smoothness problem.
During the effort, the AI agents exchanged nearly 3 million messages and generated 130 billion output tokens, referring to individual units of text and code produced in responses.
Based on OpenAI’s commercial pricing for its most advanced models, the computing effort would have cost about $10 million (£7.3 million).
The work addressed two of the four statements required for the proof set out by the Clay Mathematics Institute, the US-based organisation that oversees the Millennium Prize. The prize offers $1 million to the first party to solve designated mathematical problems.
However, OpenAI’s solution has not yet been independently verified or publicly accepted by the institute.
OpenAI said on Tuesday that it was publishing the result to highlight the substantial progress made by its AI models and confirmed that it did not intend to claim the $1 million Millennium Prize.
Allegations over timing and methods
The claim has nevertheless sparked controversy among mathematicians.
Hours before OpenAI published its work on Tuesday, Tristan Buckmaster, a mathematics professor at New York University, issued a public statement questioning the timing and methods behind the company’s work.
Buckmaster and Levent Alpöge, a mathematician at OpenAI rival Anthropic, had also been working on solutions to the Navier–Stokes problem using OpenAI’s Codex software tool.
Buckmaster said he learned on 3 September that information about their progress had been passed to OpenAI. He alleged that OpenAI only began working on the Navier–Stokes equations after information about their ongoing research had reached the company.
He also shared text from email exchanges with OpenAI concerning the matter.
Buckmaster said he had not yet read OpenAI’s complete proof but felt compelled to make the information public regarding what he had been told, when he had been told it and what had been proposed.
He said his aim was to ensure that an incorrect sequence of announcements was not accepted as fact.
OpenAI responds
OpenAI on Tuesday congratulated Buckmaster and Alpöge on their concurrent work, describing their research as “remarkable”.
The company maintained that it had not seen any of their work through any medium until it was publicly released. It also said no user data had been accessed during its Navier–Stokes project.
However, OpenAI acknowledged that it could not completely exclude the possibility that information derived from the researchers’ use of its products had contributed to improvements in its models.
“While unlikely, we cannot rule out that de-identified data derived from their usage of our products helped improve our models,” OpenAI said.
The company stressed that its mathematical proofs differ significantly from those produced by Buckmaster and Alpöge, adding that the precise results proved by the two efforts are also distinct.
