OpenAI said on Tuesday that its newest artificial intelligence technology had solved one of the “Millennium Problems,” a collection of important unanswered math questions meant to push the world’s leading mathematicians to new heights.
The announcement is an another clear sign that A.I. is fundamentally changing the upper reaches of mathematics, which have long been viewed as a pinnacle of human achievement. The change has excited some mathematicians, while stirring concern among others.
Over the past year, A.I. systems successfully solved a wide range of problems that have bedeviled mathematicians for decades. But these problems were not as complex, nor as closely watched, as the one that OpenAI’s technology has solved over the past several days. The Millennium Problems are among the most heavily researched in the field.
“This is a spectacular culmination of the arc we have seen over the past twelve months,” OpenAI researcher Sebastian 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 liquids. 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.”
But as he and others point out, mathematicians still provide a helping hand as A.I. systems work through these problems. OpenAI said that it deployed vast teams of “A.I. agents” to solve the Navier-Stokes problem and that its researchers passed key ideas between these teams.
“Our role was like a bumble bee cross-pollinating across different groups and delivering different bits of information,” said OpenAI researcher Dan Roberts.
A.I. systems may also draw on key breakthroughs previously made by top mathematicians — or independently duplicate their work. The night before OpenAI made its announcement, Tristan Buckmaster, a professor of mathematics at New York University, said in a post to social media that he had been exploring similar research with another mathematician who works for OpenAI’s chief rival, Anthropic.
In a blog post on Tuesday, OpenAI acknowledged that it focused its resources on the Navier-Stokes problem after hearing that other mathematicians were exploring similar research. But the company said it “did not see any of their work through any means.”
Companies like OpenAI build their A.I. technologies using what scientists call neural networks, systems that learn skills by analyzing vast amounts of digital data. About two years ago, such companies started to hone these systems using another technique called reinforcement learning. Through this process, A.I. systems can learn additional behavior through extensive trial and error.
By working through thousands of math problems, for instance, they can learn which methods lead to the right answer and which do not. Researchers inside labs like OpenAI develop complex feedback mechanisms that show the system when it has done something right and when it has done something wrong.
(The New York Times has sued OpenAI and Microsoft, claiming copyright infringement of news content related to A.I. systems. The two companies have denied the suit’s claims.)
Reinforcement learning is difficult to perfect in areas like creative writing, philosophy and ethics, where the question of which answers are right or wrong is hard to define objectively. But the technique is ideally suited to mathematics.
Through this process, A.I. systems can learn to prove mathematical theorems using a computer programming language called Lean. The language was originally designed as a tool for human mathematicians. But now that A.I. systems are skillful enough to generate their own computer code, they, too, can use Lean to generate their own math proofs.
In January, OpenAI and another start-up called Harmonic said that together, two of their A.I. technologies had solved one of the “Erdos problems,” a collection of thorny unsolved problems posed by a 20th-century academic named Paul Erdos.
Some mathematicians pointed out that the solution generated by the A.I. systems had not been all that different from earlier work done without help from A.I. “It feels to me like a really clever student who has memorized everything for the test but doesn’t have a deep understanding of the concept,” Dr. Tao told The Times.
But as the months passed, technologies from OpenAI and other companies continued to crack additional problems, sometimes in more impressive ways.
OpenAI said that it solved the Navier-Stokes problem using as many as 10,000 “A.I. agents” working in concert. Running such a large number of A.I. systems is likely to have cost millions of dollars, a result of the enormous amounts of electrical power needed to operate the specialized chips that drive A.I. technologies.
OpenAI research scientist Noam Brown called this a “very expensive process,” before adding that the costs of running A.I. technologies tend to drop as companies like OpenAI improve their efficiency.
On Tuesday, the company released a paper describing its solution, including a Lean proof, allowing outside mathematicians to see how its agents cracked the problem. But to Dr. Tao, it is still a poor substitute for humans solving the problem on their own. He compared OpenAI to a wilderness guide who finds a path to a hidden waterfall.
“This has some value,” he said. “But once someone shows a specific path to the waterfall, people just take that path. They don’t spend as much time looking for other paths.”
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