OpenAI Claims to Solve Navier-Stokes Equations: Breakthrough or Controversy?

Recently, the AI science channel Two Minute Papers released a video that sparked widespread attention, claiming that OpenAI used an AI system more powerful than its publicly available products to likely solve the Navier-Stokes existence and smoothness problem — one of the seven Millennium Prize Problems carrying a one-million-dollar bounty from the Clay Mathematics Institute, a challenge that has stumped mathematicians for over a century. The video's creator, Károly Zsolnai-Fehér, who spent several years conducting research and writing papers in the field, admitted in the video: "I didn't expect to see this problem solved in my lifetime, and now it's very likely been solved — I'm sitting here in disbelief."

That said, this alleged "historic breakthrough" has not come without controversy. This article breaks down the technical background of the Navier-Stokes equations, the direction of OpenAI's proposed solution, and the data ethics debate surrounding it.

What Are the Navier-Stokes Equations?

The Navier-Stokes equations are fundamental partial differential equations that describe fluid motion. The turbulence of rivers and the diffusion of smoke may seem hopelessly complex, but all of it can be understood through three core terms.

Advection — If you throw an object into a river, it gets carried along by the current. That's advection. The tricky part is that fluids also "advect themselves," and this term — described by a directional derivative in the equations — is the central challenge of fluid simulation.

Pressure — Think of a crowded bus: high-density areas push against each other, creating an outward tendency of motion.

Diffusion — Differences get averaged out over time. Drop a single drop of ink into water, and while its position is clearly visible at first, within a few minutes the entire glass is uniformly colored. That's diffusion.

Beyond these, fluids also respond to external forces (a puff of air sets them in motion) and are constrained by the incompressibility condition — the volume of a fluid is conserved over time; it cannot appear or disappear out of nowhere. Discretizing these equations onto a grid allows us to write computer programs that simulate the real world, with applications ranging from liquid simulation to wind tunnel testing for aircraft.

The Core Problem: Can the Math Break Down?

So what exactly is this million-dollar Millennium Prize Problem asking?

The core question is: If you start from a smooth initial flow state and let the Navier-Stokes equations run indefinitely, can the solution be guaranteed to always "behave well" and remain smooth? Or will it develop pathological behavior — such as infinities — in finite time?

According to the video, OpenAI's answer is: it breaks down. The Navier-Stokes equations do not guarantee that solutions remain well-behaved forever.

Specifically, OpenAI's approach starts from a state of rest and uses carefully constructed external forces to produce an inward-spiraling vortex. This vortex is continuously stretched, causing its velocity to grow without bound in finite time, while the system's total energy remains finite. When this occurs, the mathematical description provided by the equations "collapses."

Interestingly, the video's creator notes that this pathological scenario doesn't actually occur in nature. Once the system approaches this extreme state, physical effects at the molecular level take over, and the Navier-Stokes equations themselves are no longer applicable. The creator also candidly emphasizes that he is "just a student," and that his interpretation may be imperfect — viewers should form their own judgments.

Three and a Half Days — and the Controversy That Followed

One of the most shocking details of this story is the speed involved. OpenAI didn't spend years working on this in secret — according to the video, the AI model found its solution in approximately three and a half days.

However, controversy quickly surfaced. As a research scientist, the video's creator admits this part made him "quite uncomfortable," but he felt obligated to share it honestly.

As relayed in the video, two scientists had previously made substantial progress on a related problem. According to legendary mathematician Terence Tao, their work had advanced to the point where it was "likely extendable to solve the Navier-Stokes problem." After hearing rumors about their progress, OpenAI activated this more powerful AI system and ultimately produced a complete solution.

This raises a sharp ethical question: did the AI "borrow" the ideas and data of these human researchers? OpenAI's official response was a single sentence:

"While unlikely, we cannot rule out that data obtained from their use of our products helped improve our models."

In other words, whatever users type into the chat box could potentially be used to train proprietary AI models. The video's creator uses this to repeatedly underscore a key point: if you use a free, open-weight AI system running locally, this cannot happen — your prompts never leave your own machine.

Why Is AI Advancing So Rapidly in Mathematics?

The video concludes with a thought-provoking insight: why is AI progressing so rapidly in mathematical reasoning?

The answer lies in verifiability.

The creator draws a compelling comparison: if you ask an AI to write an article in your style, you need to read and evaluate the quality yourself — perhaps a hundred articles per hour. But mathematics is entirely different: mathematical proofs can be verified automatically. A machine can check hundreds of millions of attempts per hour to determine whether a solution is correct.

"One hundred verifications per hour vs. one hundred million verifications per hour" — that's the fundamental reason AI is accelerating its breakthroughs in mathematics. This auto-verifiable property is enabling AI's mathematical capabilities to improve at a staggering pace, and it's almost certain to keep getting stronger.

The creator further cites the view of Nobel laureate and DeepMind founder Demis Hassabis: if we could transform "curing diseases" into an automatically verifiable problem like a mathematical proof, conquering more diseases within the next decade is not unimaginable. Of course, precisely because AI capabilities are growing so rapidly, the creator also calls on society to invest greater coordinated effort in AI safety and alignment.

Cautious Optimism: The Breakthrough Still Awaits Verification

Whether or not OpenAI has truly and independently solved the Navier-Stokes existence and smoothness problem, this episode reveals the enormous potential of AI in mathematical reasoning — while also exposing the accompanying issues of data ethics and intellectual property. The controversy over data sourcing and the contributions of human researchers reminds us that, even as we marvel at technological breakthroughs, we must remain vigilant about transparency and academic attribution.

This remains a result that requires rigorous scrutiny and peer review from the independent mathematical community. But one thing is already clear: using verifiability as a lever, AI is rapidly conquering mathematical and scientific heights that were once considered out of reach.