OpenAI may have claimed to have solved a million-dollar math problem that’s fundamental to lots of modern technology, but don’t expect anything to change—at least, not immediately.
Earlier this month OpenAI claimed one of its internal models had proved that the Navier-Stokes equations, which are used to model the motions of fluids, break down and yield nonsensical answers in certain scenarios. The equations can be thought of as “essentially Newton’s second law, applied to a little fluid parcel,” says Justin Beroz, CEO of ReynKo, an engineering company that specializes in turbulence.
Regardless of whether OpenAI can actually take the credit for solving the problem, the notion that an artificial intelligence has apparently succeeded where humans have, for so long, failed is earth-shattering for mathematicians. That isn’t the case for engineers, however, say several experts working in fluid dynamics.
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The Navier-Stokes equations are used to great success in a wide array of applications, such as calculating airflow over a jet’s wings, forecasting weather patterns and determining how blood flows through biomedical devices. Even so, mathematicians have spent nearly 200 years struggling to prove that the models fully correspond with reality. At very small scales, they have suspected, the equations might predict something physically impossible: fluids that accelerate to infinite speeds.
Still, as to how much this affects engineering, Beroz says, in short, “not much.”
“The longer answer is a bit more nuanced,” he says.
The knowledge that the equations break down at a certain point doesn’t really change how they will be used because the equations themselves are only an approximation of how a fluid will react, says Florian Schäfer, an assistant professor at New York University. What really matters, he says, is that this approximation is good enough that any flaws are trivial.
“The Navier-Stokes equations have a certain regime of validity in which we think that they're basically enough at describing what the physics will do,” Schäfer says.
OpenAI’s proof does provide an example in which the equations allow a fluid to hit an infinite velocity, but the scenario is so contrived that it’s almost certainly unlikely to have any direct relevance to any physical system in the real world. The methods the large language model (LLM) marshal,ed to achieve its result could in principle, offer fresh insights for engineers and mathematicians alike, says Spencer Bryngelson, an adjunct professor at the Georgia Institute of Technology.
Proving a breakdown in Navier-Stokes could “have many knock-on effects that are useful, even though it’s unclear sometimes how those knock-on effects will happen,” Bryngelson says.
One place where those effects could eventually be felt is in improved computer simulations, Beroz says.
“We have plenty of things where we deal with drag and lift and design these things, but the key thing is: it’s always done experimentally,” he says, because the mathematics of fluid dynamics are notoriously complicated. “We don’t have good computational tools to simulate these things in advance, and this really separates fluid engineering from other kinds of engineering.”
As an example, Beroz points to aerospace design, in which the development cycle of a new airplane can take a decade and cost billions of dollars.
“The kicker is that roughly half the time and half the cost is spent building, testing or finding physical prototypes in the wind tunnel—and the only reason you’re doing that is that the software sucks,” he says.
The upshot is that, while any formal proof of a breakdown in Navier-Stokes may not have any immediate effects on engineers, it’s hard to predict how this knowledge will be implemented in the future. Basic scientific research can lead to dramatic and unforeseen applications and technologies years down the line.
“This is clearly an example where the equations do not do what the physics would do, and then that might guide us to a new way of understanding in what sense and in what regimes these equations are incomplete,” Schäfer says.