Last week, OpenAI surprised the world by claiming that its cutting-edge artificial-intelligence model had solved the Millennium Sweepstakes problem. The Millennium Sweepstakes problem is one of a small number of notoriously difficult mathematical puzzles associated with a prize of $1 million.

According to the claim, the AI solution mathematically proved that the Navier–Stokes equations — a famous 19th-century framework for modelling fluid flow — can produce “singularities” and predict physically impossible infinite velocities in gases and liquids.

The announcement has prompted debate about AI’s growing mathematical capabilities, whether AI systems might absorb unpublished work by other researchers and who should receive credit for a mathematical breakthrough. It also raises a broader question: what could the claimed result mean for fluid-flow research?

How vortices expose the limits of Navier–Stokes equations

The solution posted by OpenAI, based in San Francisco, California, explores situations in which fluid vortices stretch until they become extremely long and thin. George Karniadakis, an applied mathematician at Brown University in Providence, Rhode Island, calculated that in air, the singularity appears when the vortex is about 70 nanometres wide.

That width is roughly the typical distance an air molecule travels before colliding with another molecule. Karniadakis said it therefore makes sense that this could be the point at which the equations break down.

The Navier–Stokes equations simplify fluid modelling by treating a fluid as a continuous material rather than as a chaotic collection of individual molecules. When only a small number of molecules are present in a given sample, that assumption is no longer a reliable approximation of reality.

The equations have a limited range

Researchers already knew that the Navier–Stokes equations are insufficient for modelling reality in some circumstances. The Millennium problem focused on incompressible fluids, which provide a good representation of liquids such as water.

Fields Medal-winning mathematician Charles Pfefferman of Princeton University in New Jersey said researchers already knew that, for compressible fluids, the corresponding equations could lead to singularities.

The equations are not suitable for dilute gases. As a result, they do not adequately explain what happens when a spacecraft re-enters the upper atmosphere or when very small amounts of fluid move through microscopic channels.

Could the Boltzmann equation offer an alternative?

One alternative is to use the Boltzmann equation. Rather than treating a fluid as a continuous substance, this approach models gases as collections of individual molecules and describes their behaviour statistically.

Yu Deng, a mathematician at the University of Chicago in Illinois who won the Fields Medal this year for work on the Boltzmann equation, told Nature in July that it remains unclear what the collapse of the Navier–Stokes equations would mean for the Boltzmann equation.

In principle, Deng said, the Boltzmann equation could also break down in certain situations. “Our understanding is very limited,” he said.

Why molecular simulations are difficult to scale

Another way to model fluids is to simulate the behaviour of individual molecules using powerful computers. This approach can be computationally expensive, however, and becomes impractical beyond the microscopic scale.

In 2024, researchers used supercomputers to simulate a record-breaking 155 billion water molecules. Even so, simulations of this kind are typically limited to systems that fit within a cube measuring just one micrometre on each side.

A “triple-decker” model for fluid flow

Some researchers address the computational challenge by dividing the problem into different scales. They use molecular dynamics to model the smallest scales, Navier–Stokes equations to describe larger scales and another set of equations to average the behaviour of groups of molecules.

Karniadakis calls this a “triple-decker” approach. It combines detailed molecular modelling with larger-scale fluid equations, potentially offering a way to study situations in which no single mathematical model is sufficient.