OpenAI says it solved the 90-year-old Navier-Stokes math problem in 88 hours

OpenAI, the company behind ChatGPT, has announced that a system of 10,000 coordinated artificial intelligence "agents" managed to solve the Navier-Stokes equation, one of the so-called Millennium Problems and a mathematical puzzle that has gone unsolved for 90 years. According to the statement published by the company, the process began on September 1 and the agents reached their solution on Saturday, September 5, that is, some 88 hours after the first agents were launched.

The Navier-Stokes equation describes the motion of fluids and is one of the seven Millennium Problems established in 2000 by the Clay Mathematics Institute, based in Cambridge, Massachusetts, which offers a one-million-dollar prize for the solution to each of them. Other problems on the list include the Birch and Swinnerton-Dyer Conjecture, related to elliptic curves, and the Riemann Hypothesis, linked to prime numbers. The institute explains that these prizes were conceived to highlight to the public that, in mathematics, the frontier of knowledge remains open and full of significant unsolved problems. So far, the Clay Mathematics Institute has not commented on OpenAI's proposed solution, so formal validation of the result is still pending.

According to OpenAI itself, the agents used had tools such as access to a cached version of the internet and the ability to run code. These agents were organized into groups able to communicate internally, of varying sizes, and the group that produced the Navier-Stokes solution came to operate with on the order of 10,000 agents running concurrently, all of them powered by an internal AI model of the company.

The announcement, however, has not been free of controversy. Tristan Buckmaster, a professor of mathematics at New York University, publicly questioned the achievement in a statement published on his own website. Buckmaster explained that he had been collaborating personally for some time with mathematician Levent Alpöge —who works at Anthropic, an OpenAI rival— on several mathematical problems, among them Navier-Stokes. As he told it, Alpöge received "warnings" that information about their progress had reached OpenAI. Buckmaster noted that the route OpenAI used to arrive at the solution resembled his own work, and said that direction "is not one you get to in a few days simply by giving a model the statement of the problem".

The professor also raised doubts about whether OpenAI's models had been trained on the sessions he and Alpöge held in Codex, OpenAI's coding tool, or had had access to them, given that both mathematicians had used several language models as part of their work. Buckmaster was careful to qualify the scope of his claims: he said he had not seen OpenAI's proof, did not know what the company's model did or how, and did not know whether his own data had in fact been used.

For its part, OpenAI responded in its statement that its effort to solve the problem started on September 1 after "hearing a rumor" about progress on the problem, a rumor they later linked to the work of Alpöge and Buckmaster. The company maintained that neither its researchers nor its agents ever saw the two mathematicians' work before it became public, and stressed that, in particular, no specific user data was accessed to solve the problem. OpenAI did qualify, however, that while it considered it unlikely, it could not completely rule out that de-identified data derived from the mathematicians' use of its products had contributed to improving its models.

The episode illustrates several underlying tensions in the current race between AI labs. On one hand, there is the interest of companies like OpenAI in demonstrating that their agent systems are capable of generating far-reaching scientific and mathematical advances, something of enormous propaganda and commercial value. On the other, there are the difficulties of independently verifying announcements of this kind, both as regards the validity of the mathematical proof itself —which must still pass the scrutiny of institutions such as the Clay Mathematics Institute— and as regards the real provenance of the knowledge that made the result possible. The doubts raised by Buckmaster point to a broader problem: the difficulty of distinguishing between a genuinely autonomous advance by an AI system and one that may have benefited, directly or indirectly, from unacknowledged prior human work, in a context where the boundaries between researchers' use of AI tools and the training of those same AIs are increasingly hard to draw.