Did OpenAI solve the wrong Navier-Stokes problem?

# Did OpenAI solve the wrong Navier-Stokes problem? Source: [https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/](https://www.scientificamerican.com/article/did-openai-solve-the-wrong-navier-stokes-problem/) Two weeks ago, OpenAI[claimed a solution](https://www.scientificamerican.com/article/openai-claims-blockbuster-math-breakthrough-amid-swirl-of-controversy/)to one of the[biggest open problems in math](https://www.scientificamerican.com/article/ai-may-have-just-solved-a-million-dollar-math-problem-the-field-will-never-be-the-same/)—the[Navier\-Stokes problem](https://www.scientificamerican.com/article/humans-and-ai-race-to-blow-up-maths-toughest-equations/), worth a million\-dollar prize from the Clay Mathematics Institute\. The proof ignited a[powder keg](https://www.scientificamerican.com/article/25-winners-of-maths-nobel-prize-decry-the-ai-invasion-of-their-discipline/)of concern over artificial intelligence companies’[race to disrupt](https://www.scientificamerican.com/article/mathematicians-confront-the-ai-apocalypse/)the subject\. But with the dust still far from settled, a different controversy is emerging: did OpenAI even solve the*right*Navier\-Stokes problem at all? 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\. --- ## On supporting science journalism If you're enjoying this article, consider supporting our award\-winning journalism by[subscribing](https://www.scientificamerican.com/getsciam/)\. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today\. --- “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, on Friday, three mathematicians[posted a proof of their own](https://arxiv.org/html/2609.20803v1)showing 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,” meaning 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 more pure, more fundamental way for the equations to blow up, 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 Luis Martínez\-Zoroa\. In the last few years, though, Córdoba and Martínez\-Zoroa have put all their focus on this niche piece of the equations, laying out a method to build a very specific external force that triggers blowup\. On September 7th, two mathematicians used their program \(and AI\) to produce blowup for a frictionless fluid—considered a major step toward blowing up Navier\-Stokes\. Less than a day later—timing which has led to a heated dispute between the groups—OpenAI finished the job\. But the new work from Friday 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 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,” says Silvestre\. 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](https://www.claymath.org/wp-content/uploads/2022/06/navierstokes.pdf), penned in 2000 by the mathematician Charles Fefferman, offers an option called “C” where solutions are allowed to use an external force like OpenAI’s\. Now, fluid dynamicists are grappling with a possibility few had considered before: That the Navier\-Stokes equations can blow up, but*only*with an external force\. You can mathematically place a fluid in a specific, unrealistic situation to break the equations, but 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 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 it 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\.”