OpenAI Claims AI Solved the Navier–Stokes Problem With Thousands of Agents

From AI-generated music to mathematical proofs, artificial intelligence has shaken up the creative industry—and now mathematics. OpenAI announced that it had solved a decades-old puzzle known as the Navier–Stokes existence and smoothness problem using thousands of AI agents.

“Artists and musicians have already experienced this,” says Jaspreet Singh Sandhu, a mathematician at Colorado State University.

Why AI’s Approach to Mathematics Is So Unusual

Despite the time constraints of high school exams, mathematicians are not primarily focused on finding the correct answer as quickly as possible. Contrary to the image of mathematics as a practical subject, developing new mathematical ideas often resembles an artistic pursuit—similar to inventing games or solving puzzles.

Mathematicians typically follow a thoughtful and careful process as they develop their ideas. OpenAI, by contrast, approached the proof through brute force, shortcutting the process in ways that could undermine human understanding.

Mathematicians, for example, build patterns from ideas. “Painters and poets are makers of patterns,” the British mathematician G. H. Hardy wrote in his 1940 essay A Mathematician’s Apology. Painters use shapes and colors, poets use words, and mathematicians construct patterns from mathematical ideas.

Hardy’s Case for “Useless” Mathematics

Hardy, a pacifist, wrote the essay during World War II to argue that people should pursue mathematics for its own sake, apart from its applications—especially during wartime. His essay promotes the value of “useless” mathematics. He quotes the famous mathematician Carl Friedrich Gauss’s oft-cited remark about number theory, the branch of mathematics that studies whole numbers.

Gauss’s remark epitomizes the idea of beautiful but useless mathematics. Although Hardy’s comparison between mathematics and art holds, he was wrong about some of the details. After centuries of being considered useless, number theory became valuable through cryptographic protocols that are now widely used to secure emails and bank accounts.

What Is the Navier–Stokes Problem?

For now, however, uselessness is exactly what OpenAI has proven. The puzzle it claims to solve is one that has intrigued mathematicians for decades.

The problem takes its name from the Navier–Stokes equations, developed by 19th-century scientists to describe the flow of viscous fluids. Engineers use these equations to model airflow when designing airplanes. Mathematicians, however, became fascinated by the equations themselves rather than their engineering applications.

“Mathematicians’ primary interest in equations was certainly not engineering,” said Vanderbilt University mathematician Jared Speck. They pursued answers for “the mathematical richness, the puzzle aspect,” he says.

Solving the problem will not help engineers design a more aerodynamic airplane wing. In other words, many cakes are cylindrical, but studying the equations that describe cylindrical shapes will not necessarily help you bake better cakes.

Why Mathematicians Find the Puzzle Compelling

“When a problem can’t be solved, it has a bit of a lore attached to it,” Speck says. The appeal of the Navier–Stokes problem is similar to the appeal of Sudoku or chess, except that it involves a mathematical equation describing the flow of fluids in the world around us.

Mathematicians imagined how the equation might behave in unusual, almost science-fiction scenarios. They wanted to know whether the equation implied that a fluid could explode for no physical reason under unrealistic conditions.

Mathematicians hope that approximate equations such as the Navier–Stokes equations can reveal seemingly nonsensical situations. They find these possibilities especially interesting because they can lead to entirely new mathematical ideas.

According to Speck, the mathematical community has developed “deep and beautiful theories” around the equation for years. Researchers were close to solving the problem until OpenAI’s proof showed that the Navier–Stokes equations did, indeed, imply a science-fiction-like fluid explosion.