Summary
OpenAI says its most advanced AI model produced a mathematical result involving a singularity in the Navier–Stokes equations. The case highlights why continuum fluid models need molecular descriptions at very small scales.
OpenAI says its most advanced artificial-intelligence model has produced a mathematical result concerning the Navier–Stokes equations, a set of nineteenth-century equations used to describe fluid motion. The claimed result identifies a situation in which the equations develop a singularity: a point where the mathematics predicts an infinite value, such as infinitely high speed.
That is not a prediction that air or water will physically reach infinite speed. It is a sign that the mathematical model has been pushed into a regime where its assumptions no longer provide a realistic description of the fluid.
A vortex stretched to molecular scale
The scenario described by OpenAI involves a vortex that stretches until it becomes extremely long and thin. In a calculation by applied mathematician George Karniadakis, the singularity appears for air when the vortex is approximately 70 nanometres wide.
That scale is significant because it is roughly the typical distance an air molecule travels before colliding with another molecule. The Navier–Stokes equations treat a fluid as a continuous substance. At ordinary scales, this approximation is highly useful: the motion of vast numbers of molecules can be represented by smooth quantities such as velocity, pressure and density.
When only a few molecules span the region being modelled, however, the continuous-fluid assumption becomes a poor approximation. Individual molecular movements and collisions become important. The singularity in the described case therefore sits close to a physical boundary between two ways of describing matter: fluid mechanics on one side and molecular or statistical mechanics on the other.
Why one set of equations cannot cover every scale
The Millennium Problem addressed incompressible fluids, which provide a good approximation for liquids such as water. Its scope was narrower than the behaviour of all gases and liquids. For compressible fluids, the corresponding equations were already known to be capable of producing singularities under some conditions.
The Navier–Stokes framework also becomes unsuitable for rarefied gases, where molecules are spaced far enough apart that collisions and individual trajectories matter. Examples include parts of a spacecraft’s re-entry through the upper atmosphere and very small amounts of gas moving through microscopic channels.
One alternative is the Boltzmann equation. Rather than treating a gas as a continuous medium, it models the statistical behaviour of individual molecules. That approach may provide a better description in rarefied conditions, although mathematicians quoted in the account say it is not yet clear whether the Boltzmann equation could also fail in particular circumstances.
A more direct option is to simulate molecules individually. This is computationally expensive. In 2024, a supercomputer simulation modelled a record 155 billion water molecules, a quantity that would generally fit inside a cube only a few micrometres across. Extending that approach to larger objects or longer periods quickly becomes impractical.
Researchers therefore sometimes combine methods. Molecular dynamics can represent the smallest scales, Navier–Stokes equations can handle larger scales, and an intermediate set of equations can average molecular behaviour between them. Karniadakis has described this multiscale strategy as a “triple decker” approach.
The significance of the OpenAI claim is consequently broader than whether an AI system can produce a difficult mathematical proof. It draws attention to the assumptions built into a foundational model of fluid flow. Navier–Stokes equations remain powerful for the scales and conditions where their continuum approximation works, while molecular descriptions become necessary as flows approach the scale of individual particles.