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OpenAI Says It Has Cracked One of Math’s ‘Millennium Problems’

“This is a spectacular culmination of the arc we have seen over the past twelve months,” OpenAI researcher Sébastien Bubeck said of the company’s new solution.

The company announced that one of its latest models, which has not yet been released to the public, needed just 88 hours to solve what mathematicians call “the Navier–Stokes existence and smoothness problem.” This problem involves a series of equations that are often used to predict the weather.

The equations describe the movement of water and other fluids. The Navier-Stokes problem, which has no clear practical value, asks whether these equations completely break down in certain situations. OpenAI’s proof claims to have defined just such a situation.

This would imply, at least theoretically, that the laws of physics themselves would break down under certain conditions: that, for example, water could be made to spontaneously explode. But mathematicians and physicists do not believe that this mathematical breakdown could really lead to such an outcome in the physical world.

The Navier-Stokes problem was one of seven “Millennium Problems” selected by the Clay Mathematics Institute in the year 2000 as a way of tracking the progress of mathematics in the new millennium. The institute, founded by an American businessman named Landon T. Clay, offered a million dollars for the first correct solution to each problem. Before OpenAI’s announcement, only one of the problems had been solved.

“These questions are lighthouses,” said Terence Tao, a professor at the University of California, Los Angeles, who is regarded by many as the finest mathematician of his generation. “They are great focus points that attract the efforts of human scientists.”

Dr. Tao is among the many mathematicians who have publicly warned that the latest A.I. systems could end up damaging the field of mathematics. If A.I. technologies can solve the most difficult problems with little input from human mathematicians, he says, they could weaken human understanding of the field.

“The effort needed to solve problems is often very instructive. It teaches you something. It’s like going to the gym and having a goal to lift a weight a hundred times,” he explained. “Now, A.I. can solve questions without really getting any value out of them. It’s like having machines that can lift weights for you at the gym.”