On Tuesday, September 8th, OpenAI announced that their undisclosed internal AI model had partially cracked the long-standing “Navier-Stokes equations,” a problem that has troubled humankind for nearly a century, after consuming millions of dollars worth of computing resources. However, this claim immediately sparked controversy in the academic community and among competitors regarding the primacy of achievements. OpenAI also explicitly stated that they have no intention of seeking the $1 million prize offered by the Clay Mathematics Institute.
According to a statement released by the OpenAI research team, they utilized a more powerful internal AI model than the GPT-6 Astra model, deploying around 10,000 AI agents (autonomous programs) over a span of approximately 88 hours to prove the “Navier-Stokes” equations.
The statement also mentioned that during the process, the AI agents exchanged about 2.7 million messages, consumed around 300 billion output tokens, and additionally spent 17 hours completing rigorous formal verification through GPT-6 Astra.
Mark Chen, Chief Research Officer at OpenAI, stated that the computational cost of this AI endeavor was “in the order of millions of dollars,” dubbing it a significant milestone in AI research, aiming for further breakthroughs in the future.
Researcher Sebastien Bubeck from OpenAI added that the computational cost incurred in this project was a thousand times higher than previous efforts in solving mathematical problems.
The statement from OpenAI immediately provoked controversy. Just a few hours before OpenAI’s announcement, New York University Professor and mathematician Tristan Buckmaster, along with researcher Levent Alpoge from Anthropic, published three papers demonstrating the use of AI to solve simplified versions of fluid equations, including the forced Euler equation.
Following this, Buckmaster publicly questioned the research findings of OpenAI. He pointed out that during their research, the results were stored in OpenAI’s Codex tool used for coding, implying that OpenAI might have accelerated their breakthrough upon learning about their research.
He emphasized, “I’m not accusing anyone, but I don’t know what their AI did, how they did it, and whether they utilized our data achievements.”
Facing these doubts, Bubeck publicly denied that OpenAI had accessed or utilized the research findings of this duo or data shared with OpenAI servers, nor had they employed their prompt words.
On Wednesday, September 9th, in a statement on the social platform X, OpenAI cautiously mentioned, “While it’s unlikely, we cannot rule out the possibility that data generated by users using our products may help improve our models.”
OpenAI also stressed that there are significant differences in the proof methods between the two parties, especially in the specific results demonstrated on the Euler equations (both forced and unforced versions).
The Euler equation in fluid mechanics considers fluids like water and air to be inviscid, serving as a simplified version of the Navier-Stokes equations, which account for the viscosity of fluids.
The “Navier-Stokes” equations were one of the seven Millennium Prize Problems announced by the Clay Mathematics Institute in the year 2000, with six remaining unsolved, each offering a prize of up to $1 million upon resolution.
Since its proposal in the 19th century, the “Navier-Stokes” equations have been fundamental in human research fields such as aircraft design, weather forecasting, and blood flow dynamics. This set of equations, which views fluids as a continuous medium, has been under exploration in the physics community regarding its permanent applicability under extreme conditions.
OpenAI’s research indicates that under specific conditions, these equations can “break down” over time, transforming initially normal smooth flow into rapidly rotating and elongating vortices that ultimately reach infinite speeds within a finite time, rendering the equations unable to completely reflect physical reality. This proof has been formally verified through the Lean programming language.
As per the award rules, OpenAI’s public proof must first be published in a scientific journal and confirmed by the mathematical community for two years before the Clay Mathematics Institute can convene a committee for evaluation, and upon approval, the prize money will be disbursed.
Professor Martin Bridson, Director of the Clay Mathematics Institute, expressed excitement at the announcement of significant progress in humanity’s understanding of mathematics. He also emphasized, “Our evaluation process cannot be rushed, ensuring the utmost rigor in the review process.”
OpenAI explicitly stated that they do not plan to apply for this prize money, viewing this result solely as a milestone showcasing rapid progress of AI models.