Mathematicians and scientists are reacting to a major development involving artificial intelligence and one of the most challenging problems in mathematics. OpenAI announced on September 8 that it had found a solution to the Navier–Stokes existence and smoothness problem, a challenge that has stumped experts for decades. This problem is one of seven Millennium Prize Problems selected by the Clay Mathematics Institute in 2000, each carrying a $1 million reward for resolution. The Navier–Stokes equations describe how fluids behave under pressure, density, and velocity changes over time. These equations are essential for weather forecasting, ocean studies, and engineering design like aircraft and pumps. Despite being over a century old, the full behavior of these equations has not been clearly understood, especially whether they always produce stable results.
Before OpenAI’s announcement, mathematicians Tristan Buckmaster and Levent Alpöge had made progress on a similar problem called the Euler equations, according to Science News. The Euler equations are simpler than Navier–Stokes because they do not include viscosity, which is a measure of fluid resistance to flow. Buckmaster and Alpöge worked with AI tools, including models from OpenAI, to advance their research. Rumors quickly spread about their work, prompting OpenAI researchers to begin working on the Navier–Stokes problem themselves. OpenAI used around 10,000 AI agents simultaneously to explore possible solutions. These agents discovered a scenario where a vortex becomes increasingly thin and fast until its speed approaches infinity. If the proof withstands further scrutiny, this finding would show that the Navier–Stokes equations are not always well-behaved, answering a long-standing question in mathematics.
The computational cost of running so many AI agents was estimated to be millions of dollars. OpenAI’s solution has been verified by Lean, a tool used for checking complex mathematical proofs. However, the paper detailing the work is 166 pages long, and mathematicians are still working through its contents. Mathematical physicist Gregory Eyink of Johns Hopkins University noted that no one has fully verified the proof yet from a human perspective.
This achievement is part of a broader trend showing how AI is transforming mathematics and scientific research. In the past year, AI has enabled several major discoveries in math, prompting experts to reconsider their field’s direction. OpenAI researcher Sébastien Bubeck called this result a significant milestone in the evolution of AI and mathematics.
Buckmaster compared their progress to the historic moment when Deep Blue beat chess champion Garry Kasparov in 1997. He emphasized that the mathematics community needs time to discuss what comes next. Mathematicians are indeed taking notice of this development, with some discussing it even while caring for newborns.
Despite the importance of the problem, Eyink said that the solution does not have major practical effects. The equations assume fluids act as a continuous medium, but real-world fluids are made of atoms and molecules. He added that the main value is in recognition rather than practical application.
The controversy surrounding the solution also raises questions about how credit should be shared when AI plays a role. Buckmaster said he had discussions with OpenAI researchers about how to present both sets of findings. He claimed that OpenAI requested to exclude his coauthor Alpöge, who works for a rival company called Anthropic. There were also questions about whether OpenAI’s AI agents had access to Buckmaster and Alpöge’s earlier work. OpenAI said it did not access any specific user data from their work, though it could not rule out that de-identified product usage data had helped improve its models.
Eyink said the most important takeaway may be how difficult it is to assign credit when artificial intelligence is involved in discoveries.