OpenAI Claims Breakthrough in Navier-Stokes Equation, but NYU Mathematician Alleges His Work Was Copied

Mathematician Tristan Buckmaster asserts OpenAI cribbed the work he and a collaborator had done, which included significant progress toward solving the equation in August. OpenAI announced its team had solved the equation this week.

From Legal Insurrection

Posted by Leslie Eastman

In May 2000, to inspire mathematicians and mark the start of the new millennium, the Clay Mathematics Institute in Cambridge, Massachusetts, launched its seven Millennium Prize Problems. Each challenge carried a $1 million award for a solution.

One of the seven challenges was the Navier-Stokes Equation. The Navier–Stokes equations are among the most useful mathematical tools for describing how fluids move (including liquids such as water and blood, and gases such as air). Their practical value is that they connect pressure, velocity, density, temperature, viscosity, and external forces, allowing scientists and engineers to predict or simulate flow.

Essentially, the Navier–Stokes equations express conservation of momentum for viscous fluids. The Navier–Stokes Millennium Prize Problem asked a very specific question: If you start with a perfectly smooth three-dimensional flow of an ordinary fluid, do the equations always keep producing a smooth, physically sensible flow forever, or can the flow become infinitely extreme at some finite time?

This week, OpenAI announced that it had solved the equation.

OpenAI announced September 8 a solution to the Navier–Stokes existence and smoothness problem, which is so monumental that a $1 million prize has been offered for its resolution. “It’s one of the guiding problems for the field. It is a huge deal to know the answer,” says mathematician Dallas Albritton of the University of Wisconsin–Madison, who was not involved with the new work.

However, New York University mathematician Tristan Buckmaster disputes this claim, as he and another collaborator were making meaningful progress on related work. The mathematician asserts that OpenAI copied his collaborator’s work.

…[P]rofessor Tristan Buckmaster, a British-Australian researcher who trained in Germany, has come forward alleging OpenAI may have copied his work after asking him to collaborate — with one of the company’s scientists allegedly warning he might “ruin his career” if he refused.

Buckmaster had been working with Long Island-raised math prodigy Levent Alpöge, a researcher with OpenAI rival Anthropic, on a different elusive fluid dynamics equation, known as the Euler problem, when he found a solution using AI tools from both Anthropic and ChatGPT.

He said scientists from OpenAI approached him and tried to convince him to jointly announce the solution, claiming they had independently solved Navier-Stokes.

Buckmaster questioned the timing of their finding the solution in a blog post, noting OpenAI had only done so “in the past few days” — after the company became aware of his work.

Buckmaster indicates he feels rolled over in the battle of two AI giants. It appears that when Anthropic solved one or two of math’s “Millennium Prize problems”, OpenAI’s teams were to work on the mathematical challenges themselves.

Dr. Buckmaster said he did not care about the $1 million or even winning the race to come up with the answer. “I love math, and I just wanted to be part of the story,” he said during a two-hour recounting of the events in his N.Y.U. office on Tuesday afternoon.

He is now part of a different story: the acrimonious rivalry between OpenAI and Anthropic, which was founded in 2021 by former OpenAI leaders who said OpenAI had become cavalier about the dangers of artificial intelligence.

“When they are two of the biggest companies in the world, they need to get over it,” Dr. Buckmaster said. “I am just a bystander in these petty games.”

The tension between Dr. Buckmaster and OpenAI also reflects a chasm of distrust between the swashbuckling culture of high-flying, high-tech start-ups and the usually cordial and communal collaboration of mathematicians, who nonetheless increasingly rely on the tools of A.I.

There was an attempt at a compromise, but OpenAI claims it denies that its team ever saw the pair’s work before publication.

Tristan Buckmaster, a mathematics professor at New York University, alleged that OpenAI moved on the problem only after word of his unpublished results reached the company. He had worked for nearly a year with Levent Alpöge, a mathematician employed by rival lab Anthropic.

Both men leaned on OpenAI’s Codex assistant while assembling their own proofs, and the company can train on those sessions unless users opt out.

The pair chose a narrow route through the problem that almost nobody else was pursuing, which made the timing look suspicious to Buckmaster. He says OpenAI mathematician Sébastien Bubeck proposed a compromise that would strip Alpöge of credit, then asked why he would ruin his career.

OpenAI denies that its researchers or agents saw the pair’s work before publication. It says no specific user data went into the effort. The company concedes that de-identified data drawn from product use may have improved its models, though it argues the two proofs differ significantly.

Whether the fluids are smooth or not, the human drama surrounding its alleged solution to the Navier–Stokes equations is anything but.

Whether OpenAI’s proof was truly derived independently will determine if this revelation becomes a genuine scientific milestone or a cautionary tale about an increasingly ruthless race for AI supremacy.

In the meantime, a million-dollar prize has exposed a far more troubling equation: the unintended consequences of pairing extraordinary computational power with intense corporate rivalry, devoid of the spirit of collaboration that has long been central to scientific and mathematical progress.