GoKawiilChaotic vortices the wake of an aeroplane can contribute to aerodynamic drag.Credit: Photo12/UIG/Getty
Last week, OpenAI stunned the world by claiming that its most advanced artificial-intelligence model had solved a Millennium Prize Problem: one of just a handful of notoriously tricky mathematical conundrums that each come with a US$1-million prize. The AI solution proved mathematically that there are instances in which the Navier–Stokes equations — famous, nineteenth-century formulae that model how fluids flow — produce a ‘singularity’, predicting physically impossible infinite speeds for gases or liquids.
The news has opened up discussions about the blossoming mathematical abilities of AI, whether AI models are soaking up the unpublished work being done by others, who deserves credit and more. But there’s another question, too: what does this mean for the study of fluid flow?
The solution posted by OpenAI, which is based in San Francisco, California, explores a situation in which a vortex of fluid stretches out, becoming very long and very thin. George Karniadakis, an applied mathematician at Brown University in Providence, Rhode Island, has calculated that for air, the singularity appears when this vortex becomes around 70 nanometres wide — so thin that its width spans the typical distance that one air molecule travels before hitting another one.
It makes sense that this is the point at which the equations break down, Karniadakis says. The Navier–Stokes equations make a simplifying assumption that a fluid is a continuous substance, rather than a collection of chaotic molecules. When there are only a couple of molecules across the sample, that assumption is no longer a good approximation of reality, and the equations fail.
Limited scope
The Navier–Stokes equations were already known to be insufficient to model reality under certain circumstances. The Millennium Problem focused on fluids that are incompressible, which is a very good description of liquids such as water. For compressible fluids, researchers already knew that the corresponding version of the equations could give rise to singularities, says Charles Fefferman, a Fields Medal-winning mathematician at Princeton University in New Jersey.
The equations also don’t work for rarefied gases, so they don’t describe well, for example, what happens when a spacecraft re-enters the upper atmosphere, or when tiny amounts of fluid flow through microscopic channels.
One alternative approach is to use the Boltzmann equation, which treats a gas as a collection of individual molecules and models their behaviour statistically, rather than treating a fluid as a continuous substance as the Navier–Stokes equations do. Yu Deng, a mathematician at the University of Chicago in Illinois who won a Fields Medal this year for his work on the Boltzmann equation, told Nature in July that it is unclear what a breakdown of Navier–Stokes could mean for the Boltzmann equation. In principle, those equations could also break down in specific circumstances, too, he said. “Our understanding is very limited.”
Another possibility for modelling fluids is rigorous computer simulation of the behaviour of individual molecules. But this is computationally expensive and becomes untenable beyond the microscopic scale. In 2024, for example, researchers used a supercomputer to simulate a record-breaking 155 billion water molecules — which would generally fit into a cube just micrometres in size.
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