OpenAI has solved the Navier–Stokes problem. Did it do it on its own?

Tristan Buckmaster believed he was on the verge of making the most significant discovery of his mathematical career when he learned that OpenAI, the company behind ChatGPT, was also pursuing the same problem and was making a serious attempt to solve it.

Last month, Buckmaster, a professor at New York University, and Levent Alpöge, a mathematician at Anthropic, had privately achieved a major breakthrough on one of the most difficult problems in mathematics, the Navier–Stokes problem. They hoped that their idea could be extended to yield a complete solution, perhaps within as little as a month. The Navier-Stokes problem was one of the ‘Millennium Prize Problems’ – the most significant mathematical problems, the verified solution to which guarantees a prize of one million dollars awarded by the Clay Mathematics Institute. But whilst Buckmaster and Alpöge were trying to make sense of their results, researchers at OpenAI launched a swarm of ‘agents’ to tackle all six of the Millennium Prize Problems that remain unsolved. After just a few days’ work and nearly five million messages exchanged between AI agents working independently and fulfilling different roles, OpenAI announced last Tuesday that it had solved the Navier-Stokes problem.

OpenAI’s drive to solve the Millennium Prize problems had been triggered by a rumour circulating online that Anthropic had solved at least one of them. “One of the most incredible moments in OpenAI’s history for me has been seeing all this happen over the past week,” CEO Sam Altman wrote on social media following the announcement. And it may not be over yet. “Since solving the Navier-Stokes equations, we have made substantial progress on another Millennium Prize problem. We are considering how best to share these results,” said spokesperson Laurance Fauconnet. Buckmaster and Alpöge, meanwhile, have had to come to terms with a certain degree of disappointment.

This is not exactly a story of man versus machine, like that of IBM’s Deep Blue, which defeated world chess champion Garry Kasparov in the 1990s. Buckmaster and Alpöge were already using artificial intelligence models to speed up their work. Buckmaster was working with commercial versions of ChatGPT and Claude, paid for out of his research budget, whilst Alpöge was using in-house models to which he had access as an employee of Anthropic. Buckmaster said he was somewhat astounded by the way the models seemed to have surpassed their capabilities and by how much the world might need to “rethink what mathematics is”. However, OpenAI’s announcement also raised a question: could this have been a case of the machine ‘stealing’ from the human? Buckmaster publicly speculated that OpenAI’s models might have gained unauthorised access to his inputs whilst he was working on the problem.

OpenAI denies the allegation. “Having carried out an investigation, we can say with absolute certainty that no user input after 3 July could have influenced this system in any way,” said Fauconnet. However, the incident has highlighted the clash between the fast-paced and opaque dynamics of Silicon Valley and the measured, often collaborative and well-established norms of theoretical mathematical research. Mathematicians outside OpenAI are still trying to make sense of the company’s 166-page, densely packed paper. Some of the company’s employees are mathematicians, but none possessed research-level expertise on the Navier-Stokes problem, which made them “unable to contribute significantly to the mathematical content”, according to OpenAI researcher Sebastien Bubeck. OpenAI formally verified that the solution is correct by using a programming language to check the logic step by step.

For many mathematicians, however, understanding is just as important as correctness. Under normal circumstances, a result of this significance might take one or two years to be presented and assimilated through conferences, seminars and a formal publication, said Javier Gomez-Serrano, an expert in Navier-Stokes equations at Brown University. The Clay Mathematics Institute only considers nominations once they have been published in a scientific journal for two years and have achieved “general acceptance within the global mathematical community”. “I don’t think the mathematical community will accept the text as it stands,” said Gomez-Serrano, referring to OpenAI’s solution, which he is trying to read: “It’s incomprehensible.”

The academic world is competitive, and scholars are constantly preempting one another’s new discoveries. However, the arrival of an artificial intelligence company has complicated matters, said Buckmaster. “The theme of artificial intelligence and mathematics – that was my original story,” he said. “And now the story has become: is it ethical for artificial intelligence companies to steal a march on their clients?” The Navier–Stokes equations were developed in the 19th century to describe the flow of fluids such as water and help explain phenomena such as the weather, ocean currents and the way air flows over the wings of an aeroplane. But the equations are difficult to solve and involve factors such as pressure and the thickness, or viscosity, of the fluid. Mathematicians were unsure whether the equations always had physically plausible solutions or whether there were situations in which they might cease to work. For years, Buckmaster had been searching for what mathematicians call a ‘blow-up’: an impossible scenario in which, according to the equations, a fluid would end up reaching infinite speed in a finite amount of time – like the swirl of cream in a cup of coffee that would keep spinning at infinite speed if one waited long enough. Buckmaster wanted to explore the utility of artificial intelligence. He had successfully used deep learning tools to identify good candidates for ‘blow-ups’ in simpler versions of the Navier–Stokes equations, including in a 2025 collaboration with Google DeepMind. Then, last September, Alpöge proposed that they collaborate to try to solve the Navier–Stokes problem with the help of artificial intelligence. “Essentially for ideological reasons, I fear a world in which it is companies that can lay claim to the most profound work in pure mathematics; and, given that it seems to me that only one Millennium Prize problem will be solved in the near future, I thought, somewhat on a whim, to write to you to ask whether you had ever considered working on the problem in the traditional mathematical way,” Alpöge wrote to Buckmaster. Alpöge would have worked in a personal capacity, not on behalf of Anthropic. In mid-August, the two achieved a long-sought ‘blowup’ for Euler’s equations, a related set of fluid dynamics equations that does not account for the tricky element of viscosity present in the Navier–Stokes equations. However, the first proof generated by artificial intelligence was difficult to interpret, “one of the most hideous I’ve ever read”, Buckmaster later recounted. The researchers spent weeks trying to make sense of it.

Meanwhile, rumours began to circulate on social media that Anthropic had solved one, or perhaps two, of the Millennium Prize problems. On 1 September, OpenAI decided to launch an intensive effort, using an agent system based on a model not yet released, to tackle the unsolved problems on the Millennium Prize list and a number of smaller but related issues. It took around 50 hours and 100 agents working in coordination to achieve a ‘blow-up’ of Euler’s equations, just like Buckmaster and Alpöge, but going one step further and without resorting to adding an external force to the fluid. Following that result, the company decided to focus its resources on the full Navier–Stokes problem. Around 10,000 agents were involved in this phase. They worked for around 88 hours – almost four days. On 5 September, they arrived at a solution. It took a further 17 hours to verify that it was correct: artificial intelligence had beaten Buckmaster, Alpöge and the mathematical community to the finish line.

Although it all happened at an incredible speed, the effort was staggering: around 300 billion output tokens generated by artificial intelligence, the equivalent of roughly 40 copies of the entire English-language Wikipedia. Calculated at the commercial rates of its most advanced public model, Astra, such computing power would have cost up to 15 million dollars. The company has stated that it has no intention of claiming the cash prize. It was Buckmaster who contacted OpenAI. He had heard that something was afoot and, on 3 September, whilst the company’s agents were at work, he sent an email to find out what was actually happening. Four days later, he met with Bubeck, who by then had news to share about the agents and their results. Bubeck suggested that Buckmaster could still lay claim to a share of the victory, either by using OpenAI’s computing power to arrive at his own complete solution or by taking on the role of lead author in drafting OpenAI’s result. There was, however, a problem: Alpöge’s connection to Anthropic.

Bubeck is reported to have subsequently stated on social media that he did not wish to deny credit to anyone, but that he could not see how an employee of a rival company could have been a co-author of OpenAI’s work or had access to the company’s internal systems. Recounting the exchange, Buckmaster told the Washington Post that he was not prepared to “throw Levent under the bus” and that, as a result, coordination with OpenAI had ceased. At around midnight on Labour Day, Buckmaster published an article on the results achieved with Alpöge, alongside a lengthy statement in which he recounted the sequence of events. The following morning, OpenAI announced its own findings.

If a human researcher had solved one of the Millennium Prize problems, there would undoubtedly have been some jealousy within the mathematical research community: the only other Millennium problem solved to date has been accompanied by its own share of controversy. But there would also have been a fair amount of celebration. Instead, some of the leading mathematicians signed an open letter on Friday arguing that “the objectives of artificial intelligence companies and those of the mathematical community are seriously misaligned”.

Terence Tao, one of UCLA’s leading mathematicians, wrote on social media: “We have now seen that even a rumour that someone is working on a problem can trigger a huge amount of AI-driven work aimed at sweeping it away before the original research project has had time to realise its full potential. These incentives could now lead to a decision to no longer share any promising lines of research with the wider community, reversing centuries of tradition in open science and causing serious long-term damage to the future of the field.”

According to Buckmaster, OpenAI’s account of events overlooks the work of human mathematicians who, over the years, have built up the vast body of theory that has made a solution to the Navier-Stokes equations possible. In particular, he cites two Spanish mathematicians, Diego Córdoba and Luis Martínez-Zoroa, who developed some of the techniques underlying forced ‘blow-ups’. These lines of research were promising enough to suggest that someone might solve the Navier-Stokes equations ‘in the coming years’, Martínez-Zoroa said in an interview. For Buckmaster, there is also another disturbing possibility: that OpenAI’s models may have had some form of insight into his conversations with the company’s tools, despite his having chosen not to allow his data to be used for training. The AI agents, he said, had taken “exactly the next step” that he and his collaborator would have taken, seeking a breakthrough without introducing external forces into the equations.

Initially, OpenAI had stated that neither its researchers nor its AI agents had had access to specific user data and that the company had not seen Buckmaster’s findings before they were made public. At the height of the furore sparked by Buckmaster’s allegations, the company then said it could not immediately rule out the possibility that “de-identified data derived from the use of our products may have contributed to improving our models”. However, following an investigation, OpenAI said it had ruled out this possibility regarding Buckmaster’s most recent use of the models. “We can say categorically that it is impossible for Professor Buckmaster’s Codex prompts from the last two months to have influenced the system in any way, including during training,” said Fauconnet. Buckmaster’s response was: “Quite simply, I don’t believe him.”

Miriam Waldvogel

Copyright Washington Post