Did OpenAI Solve the Wrong Navier-Stokes Problem?

Summary

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. On supporting science journalism If you're enjoying this article, consider supporting our award-winning journalism by subscribing. 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, 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 f…