OpenAI says it has solved one of math’s ‘Millennium Problems’

This AI-generated OpenAI illustration shows a snapshot of local incompressible motion. Orange marks faster angular rotation; teal marks slower rotation. Circulating speed also depends on radius. The trajectories show inward spiralling and axial stretching. (Credit: OpenAI via CNN Newsource)

OpenAI on Tuesday said its artificial intelligence technology has cracked one of the Millennium Problems –– seven famously difficult and unsolved questions in mathematics that have stumped even the world’s brightest minds.

In a release of its findings, the company said an internal model, not released to the public, solved the Navier-Stokes problem, a 200-year-old puzzle related to the motion of fluids.

The company said its model took 88 hours to solve the problem, and that it used as many as 10,000 “AI agents” working somewhat concurrently.

“This milestone represents substantial work by mathematicians and AI researchers. However, this is not a culmination, but rather a snapshot in time, of progress on AI development,” the company wrote in its statement.

Yet the notice has also raised a stir in the mathematics community for another reason.

The night before OpenAI’s announcement, New York University math professor Tristan Buckmaster released a statement saying he and Anthropic researcher Levent Alpöge had been looking into the Navier-Stokes problem –– a culmination of work that began last year – and had used a mix of large language models, including OpenAI’s Codex.

As they prepared to go public with their results, they say they reached out to OpenAI researchers –– and later learned researchers were about to make a similar discovery.

“I would like to be clear about what I am not claiming,” Buckmaster said in the statement. “I have not seen OpenAI’s proof. I do not know what their model did, or how. I do not know whether our data was used. I am not accusing anyone of anything.”

Addressing concerns OpenAI’s model could have used their work in progress to come to its own conclusion, OpenAI acknowledged that its work solving the Navier-Stokes problem began on Sept. 1 after hearing what it called a rumor of another Millennium Prize problem being resolved.

It said its agents had access to tools such as the ability to read from a cached version of the internet and the ability to run code, but that it “did not see any of their work through any means until they released it publicly” and “no specific user data was accessed in order to solve this problem.”

OpenAI said it reached out to Buckmaster and Alpöge to offer them a concurrent release of results and “visibility into all of the prompts we used and to later see the proof,” according to the Tuesday statement. Alongside its release Tuesday, OpenAI also published its Lean proof, which shows how its agents cracked the problem.

The company acknowledged, however, that: “we cannot rule out that de-identified data derived from their usage of our products helped improve our models⁠.”

Some of the world’s most highly respected mathematicians have weighed in on OpenAI’s results –– expressing concern over how using AI to solve such complex mathematical problems can risk flattening out the field.

“It’s a little like going to watch a movie and jumping straight from the first ten minutes to the last ten minutes; technically, all the plot lines are resolved, but most of the value of the experience was lost,” Terence Tao, a math professor at the University of California, Los Angeles, told CNN.

Tao acknowledged that while it’s possible OpenAI drew on ongoing efforts from human mathematicians to solve the problem, “the dynamic is now that of frenetic competition,” he said.

“The indiscriminate use of AI is turning the subject into a meaningless production quota ‘game’ that ultimately is of very little benefit, either to mathematics or to the world,” he said.

The Navier-Stokes problem, dating back to the 19th-century work of Claude-Louis Navier and George Gabriel Stokes, involves fluid dynamics and whether these equations break down under certain conditions.

The seven “Millennium Problems” were chosen by members of The Clay Mathematics Institute of Cambridge, Massachusetts, as a collection of the most important mathematical questions of the new millennium.

In 2000, the Millennium Prize Problems was established to inspire work towards solving these complex problems, offering a US$1 million prize for the first correct solution to each problem.

Only one Millennium Prize problem has been officially solved so far: the Poincaré Conjecture, a theorem about the shape of three-dimensional spaces.

In the Tuesday announcement, OpenAI said the company does not intend to claim the Millennium Prize for the result.