Two mathematicians have made major progress on the Navier-Stokes equations, one of the world's most famous unsolved mathematical problems, with substantial help from artificial intelligence. Tristan Buckmaster at New York University and Levent Alpöge at Anthropic announced three important results that edge the field closer to a complete solution worth a $1 million Millennium Prize.

What Are the Navier-Stokes Equations and Why Do They Matter?

The Navier-Stokes equations have been used to model fluid flow for over two centuries. They underpin critical real-world applications, from designing aircraft wings and simulating blood flow through arteries to building space rockets. Despite their widespread use, the equations have a troubling habit of occasionally breaking down and producing nonsensical results in certain scenarios. Solving this problem means determining whether the equations completely align with reality or whether their modeled smoothness and turbulence can deviate from actual physics.

The Navier-Stokes problem is one of six remaining Millennium Problems published by the Clay Mathematics Institute. Whoever solves it stands to win a $1 million prize, making it one of the most prestigious open questions in mathematics.

How Did AI Help Solve This Mathematical Challenge?

Buckmaster and Alpöge used large language models (LLMs), which are AI systems trained on vast amounts of text to understand and generate human language, from Anthropic and OpenAI to accelerate their work. The pair built on previous research by Diego Córdoba and Luis Martínez-Zoroa, extending it to completion with what they describe as "a great deal of help from LLMs".

The researchers achieved three key results. Two were published alongside Lean formalisation, a process that converts mathematical theories into computer code to rigorously check for logical errors. The third result awaits completion of its formalisation before publication. Importantly, their findings relate to close cousins of Navier-Stokes, specifically the Boussinesq approximation and the Euler equations, rather than the full Navier-Stokes problem itself.

Buckmaster framed the significance of this work not just in the results themselves, but in what it reveals about AI's role in mathematics. He called it "a Deep Blue-Kasparov moment," referencing the 1996 chess match where an IBM supercomputer defeated world champion Garry Kasparov, suggesting that AI is becoming a powerful amplifier of human mathematical effort.

What Do Experts Say About the Path Forward?

Leading mathematicians see the work as a crucial stepping stone toward a full solution. David Silvester at the University of Manchester noted that the paper focusing on Euler equations demonstrates that spontaneous "blow-ups" or turbulence can appear, addressing a fundamental uncertainty about the problem.

"It's a really hard problem because when it was stated, it wasn't clear whether the result was true: that is that there are smooth solutions and it stays forever stable, or, in fact, there is some blow-up," explained Silvester.

David Silvester, University of Manchester

Terence Tao at UCLA, one of the world's most respected mathematicians, expressed optimism about extending these results to the full Navier-Stokes problem. He wrote on social media that he sees no fundamental barrier to applying these methods more broadly and suggested that "pouring an enormous amount of compute and AI assistance at such a task" could yield a complete solution.

Silvester also noted that the current work alone may be sufficient for Buckmaster and Alpöge to claim a Millennium Prize, even without solving the complete Navier-Stokes problem. However, Camilla Nobili at the University of Surrey cautioned that extending these results to Navier-Stokes will require more than simply running the same technique on a more complex system, as Navier-Stokes adds friction and dissipation elements that naturally smooth out simulations.

Steps to Understanding AI's Role in Mathematical Breakthroughs

• LLM Assistance: Large language models help mathematicians by processing vast amounts of existing research, identifying patterns, and suggesting promising directions for proof development that might take humans significantly longer to discover.
• Formalisation and Verification: Converting mathematical results into computer code using systems like Lean allows AI and humans to rigorously verify that proofs contain no logical errors, increasing confidence in the results.
• Iterative Refinement: AI can help mathematicians test multiple approaches rapidly, allowing them to explore dead ends quickly and focus human effort on the most promising paths forward.

Despite the breakthrough's mathematical significance, Silvester emphasized that it may not change practical applications. Computer models of fluid dynamics have become so sophisticated that they have largely replaced wind tunnels in engineering. "Nothing will change in the applications where Navier-Stokes is used because of this result," he stated. "It's a mathematical nicety, honestly".

Silvester

The work represents a watershed moment in how mathematics is conducted. Rather than replacing human mathematicians, AI is functioning as a collaborative tool that amplifies their capabilities, allowing them to tackle problems that have resisted solution for centuries. As more mathematicians adopt similar AI-assisted approaches, the pace of progress on other famous unsolved problems may accelerate significantly.

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