Two weeks ago OpenAI claimed a solution to one of the biggest open problems in math—the Navier-Stokes problem—an achievement worth a $1-million prize from the Clay Mathematics Institute. The proof ignited a powder keg of concern over artificial intelligence companies’ race to disrupt the subject.
But with the dust still far from settled, a different controversy is emerging: Did OpenAI even solve the right Navier-Stokes problem?
Generated by an internal large language model (LLM), OpenAI’s proof relies on an approach that many experts find unnatural. It solves a variant of the problem that mathematicians say is disconnected from reality and thus less interesting. In a sense, the LLM found and exploited a loophole in the framing of the question.
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“The most important problem is unsolved,” says Luis Silvestre, a mathematician at the University of Chicago. “The Clay problem is settled, but the main problem for the Navier-Stokes equations is not.”
Furthermore, last Thursday, three mathematicians posted a proof of their own that showed that OpenAI’s method can never be extended to solve the full problem. In other words, the loophole will never be closed, barring some completely new idea.
The Navier-Stokes equations are supposed to describe how fluids flow, but mathematicians doubt whether they can always be trusted. The million-dollar problem is about whether the equations ever “blow up,” which would mean they’d allow the flow to be infinitely fast at points—something that can’t happen in the real world.
But there’s one piece of the equations that’s optional—sometimes it’s there; sometimes it isn’t. The “it” here is an external force such as gravity that affects how a fluid moves. “All fluids we know of are under some kind of external force,” says mathematician Diego Córdoba. “So to have the force makes complete sense.”
When most experts think about the Navier-Stokes problem, though, it’s without this force. They want to find a purer, more fundamental way for the equations to blow up by using only the intrinsic forces within any fluid—not by applying some specific, precisely contrived external force. “Most of the other groups, it’s true, were specifically considering the scenario without a force,” says mathematician Luis Martínez-Zoroa.
In the past few years, though, Córdoba and Martínez-Zoroa put all their focus on this niche piece of the equations and laid out a method to build a very specific external force to trigger a blowup. On September 7 two other mathematicians used their program (and AI) to produce a blowup for a frictionless fluid—considered a major step toward blowing up Navier-Stokes. OpenAI finished the job less than a day later—timing that has led to a heated dispute between the latter two mathematicians and the company.
But the new work released last Thursday shows unequivocally that if the force is removed, the blowup will disappear. OpenAI’s method can never work, in fact, without using a very contrived equation for the force that is unlike anything that could occur in the real world. “They essentially prove that the formulation with an external force was different from the problem we really wanted to solve,” Silvestre says.
In other words, the LLM’s result does not—and will never—answer the Navier-Stokes problem that mathematicians really care about.
It did, however, unambiguously solve the problem according to the Clay Institute’s original formulation. The official problem statement, penned in 2000 by mathematician Charles Fefferman, offers an option called “C,” in which solutions are allowed to use an external force like OpenAI’s.
Now fluid dynamicists are grappling with a possibility few had considered before: the idea that the Navier-Stokes equations can blow up but only with an external force. In this scenario, you can mathematically place a fluid in a specific, unrealistic situation to break the equations, yet the blowup can never come from the fluid itself.
“It’s really uncertain at this moment,” says mathematician Gonzalo Cao-Labora. “Depending on the answer to this, I think the contribution of OpenAI will be regarded differently.” If it turns out to be true, he adds, “people would probably think about the Clay problem and say, ‘We shouldn’t have put the external force in the statement.’”
This might even be good news for “team humanity.” LLMs are great at finding blowups that exist—at searching the infinite landscape of fluid scenarios and plucking out the precise situation that breaks the equations. But proving that blow-up is impossible is a kind of math AI still struggles with. “We may be at less of a disadvantage, or maybe an advantage, compared to LLMs,” says Cao-Labora. “LLMs are especially good at constructing things that are very explicit and not as good—for now—in making new theory.”
But regardless of whether the path to solving the full Navier-Stokes problem requires a clever new blowup or a whole new mathematical discipline, everyone is becoming more hesitant to bet against the machines.
“We are really amazed with how [the technology] has evolved in the last year—so we don’t know how it will look in one year,” says Cao-Labora. “It’s really a wake-up call to the community.”
