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For the first time, a truly major open problem in mathematics has been solved by computer, according to OpenAI, which says it has cracked one of the ‘Millennium Problems’ related to the motion of fluids. The artificial-intelligence (AI) company in San Francisco, California, announced on 8 September that it has generated a solution to the most commonly used physical model of fluids — the Navier-Stokes equations — showing that they can break down. Whether or not this could happen was one of the seven ‘Millennium Problems’ asked by the Clay Mathematics Institute at the turn of the twenty-first century. Solutions come with a prize of US$1 million.
“It is certainly an exciting day, as we contemplate the announcement of major advances in the human understanding of mathematics,” says mathematician Martin Bridson, president of the Clay Mathematics Institute.
“Our proof does show that there exist fluids which start out perfectly normal, and under the Navier-Stokes equations, actually achieve infinite speed in a finite amount of time,” said OpenAI computer scientist Ven Chandrasekaran in a press briefing. Because this behaviour is physically impossible for a real fluid — such as a liquid or a gas — it suggests that under certain circumstances, the equations may not be a reliable mirror of physical reality, Chandrasekaran said.
“This is, to me, the spectacular culmination of the arc we have seen over the last 12 months,” as AI solved problems of increasing complexity and importance, OpenAI mathematician Sebastian Bubeck told reporters.
The OpenAI researchers said they had been testing the ability of their latest AI prototype on all six unsolved Millennium Problems. They then decided to focus their resources on the Navier-Stokes problem on 1 September, after hearing rumours that two mathematicians, Levent Alpöge of Harvard University in Cambridge, Massachusetts, and Tristan Buckmaster of New York University, had solved some version of the fluid-motion puzzle using Anthropic AI’s models.
On 7 September, Alpöge and Buckmaster released a paper in which they say they had found a solution for the fluid equations that also achieved infinite speed, but in the simplified case in which the fluid has no viscosity. To do so, they used Anthropic AI’s Claude and OpenAI’s Codex and Astra models. They also said they had obtained a solution to the more general problem, to be released soon.
On the same day, Anima Anandkumar, a computer scientist at the California Institute of Technology in Pasadena, and her collaborators also released a solution for the zero-viscosity problem, which they obtained using a ‘physics-informed neural network’, rather than a general-purpose large language model. Bubeck told reporters that to attack the problem, OpenAI put an unprecedented amount of resources on a single task. Their model first answered a simplified version of the question in 50 hours using 1,000 AI ‘agents’. “Then we decided to go for the full Navier-Stokes, and we increased the amount of compute,” Bubeck said, putting 10,000 agents on the problem.
Terence Tao, a mathematician at the University of California in Los Angeles, described the work by Alpöge and Buckmaster as a “remarkable achievement” on the social media platform Mastodon.
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