AI Effort Tackles Complex Fluid Dynamics Problem
OpenAI has announced that its artificial intelligence (AI) has developed a solution to a highly complex, decades-old mathematical problem. The company stated that the breakthrough was achieved in a matter of hours, utilizing a newly developed AI model and deploying thousands of autonomous AI agents. According to the report, OpenAI solved the problem over an 88-hour period, marking a significant advancement in the capabilities of AI tools.
Details of the Navier–Stokes Problem
The mathematical difficulty addressed is related to the Navier–Stokes equations, which govern the movement of fluids. For ninety years, certain crucial components of this problem—specifically, a rigorous proof underlying the correct mathematical equation—had remained unsolved. OpenAI described the solution as a major “milestone” for the company, suggesting that AI tools are rapidly improving in mathematical capability.
The company explained that its journey began with training a new model toward the end of August. AI models are computational programs trained on vast datasets to identify and predict patterns. Although this new internal OpenAI model is significantly more capable than its previously released versions, researchers utilized it for several notable advanced mathematical challenges.
The effort escalated after OpenAI reportedly received rumors on September 1 that two problems associated with the Millennium Prize—a competition offering substantial rewards for solving major mathematical conundrums—had been resolved. Consequently, the company assigned approximately 10,000 AI bots trained on the new model to attempt solving the remaining problems. By September 5, roughly 88 hours later, OpenAI claimed to have found a solution for the Navier–Stokes existence and smoothness problem, which is central to understanding turbulence.
Despite the brief timeframe for the solution, the process was computationally intensive. OpenAI reported that the AI bots exchanged nearly 3 million messages and consumed 130 billion output tokens—individual lines of text or code generated by the AI model—solely for the Navier–Stokes task. Based on OpenAI’s internal pricing, this effort was estimated to cost around $10 million (or £7.3 million). The solution reportedly resolved two of the four statements required by the Millennium Prize proof for the Navier–Stokes problem, which is worth $1 million to a winner.
Our goal in releasing this result is to report on the substantial progress of our AI models. We do not intend to claim the Millennium Prize for this result.
Controversy Regarding Timing and Work Methodology
OpenAI’s announcement has generated controversy. Professor Tristan Buckmaster, a mathematics professor at New York University, publicly challenged the company’s timeline. Buckmaster stated that he and Levent Alpöge, a mathematician employed by OpenAI’s rival, Anthropic, had been independently working on solutions using OpenAI’s own tool, Codex.
Buckmaster claimed that on September 3, he learned that “information about our progress had been passed to OpenAI.” He asserted that OpenAI did not begin working on the Navier–Stokes equations until after receiving details concerning their research. Buckmaster offered to include text from exchanged emails to support his claims regarding the company’s timing and methods.
In response, OpenAI issued a statement on Tuesday congratulating the “concurrent work” of Buckmaster and Alpöge, calling it “remarkable.” The company asserted that it had not seen any of their work through any channel until it was released publicly, and confirmed that no user data was used in their work on the Navier–Stokes problem. However, the company added that, while improbable, de-identified data derived from the usage of its products might have contributed to the improvement of its models. It concluded by noting that their respective proofs differ substantially, and even the precise results proven are different.
