OpenAI says an internal artificial intelligence system has produced a solution to the Navier-Stokes existence and smoothness problem in about 88 hours, after deploying a system involving roughly 10,000 concurrent AI agents.
In a research release published on Tuesday, OpenAI said it was sharing “a solution to the Navier–Stokes existence and smoothness problem, one of the Millennium Prize Problems.”
The Navier-Stokes problem is one of seven Millennium Prize Problems established by the Clay Mathematics Institute in 2000 to highlight some of the most difficult mathematical questions that had resisted solution for years.
The OpenAI development comes after the company said its internal AI system was deployed on the problem as part of a wider effort to test its capabilities on the open Millennium Prize Problems. OpenAI said its agents reached their Navier-Stokes resolution on September 5, about 88 hours after the first agents were launched.
The company said the proof was “produced by an internal OpenAI system” and showed “that the dynamics of the Navier-Stokes equations for fluid motion can develop a singularity in finite time.”
“We’re sharing both a writeup of the proof and a formalization in Lean,” OpenAI said.
The company described the 90 year old problem as one of the “deepest questions at the frontier of mathematics”, adding that the question of whether smooth three-dimensional fluid motion can break down “has remained unresolved for roughly 90 years.”
OpenAI said its work was aimed not only at the mathematical problem but also at demonstrating the pace at which artificial intelligence systems are advancing.
“A major goal of our work is to empower scientists to advance research and technology that benefits all of humanity,” the company said.
“To solve the Navier–Stokes problem, we used an internal model that is significantly more capable than GPT-6 Astra. We believe it is important to inform the world about the pace of AI progress and what to expect from upcoming models.”
How the AI approached the problem
OpenAI said the Navier-Stokes equations use Newton’s second law of motion to describe how fluids move and are used in areas including aircraft design, weather forecasting and the study of blood flow.
It said a fundamental question has been whether the equations can develop a “singularity” even when fluid motion begins smoothly.
“Here, a singularity means the dynamics lead to speeds in the fluid growing without bound within a finite amount of time,” OpenAI said.
The company said its system produced “an analytical proof and a Lean formalization that an initially smooth fluid at rest can develop a singularity in a finite time.”
“The fluid has a smooth force applied to it, and its energy remains finite through the entire dynamics, from rest to the formation of the singularity,” it said.
OpenAI described the mathematical solution as a vortex.
“The solution is a vortex, a spinning swirl of fluid, that spirals inward and gets increasingly elongated, like spaghetti,” the company said.
“This central region shrinks while it speeds up in such a way that its energy still stays finite, as required by the laws of physics.”
The company said the technical challenge was to make the equations produce the breakdown through the movement of the fluid itself rather than by introducing an infinite force.
“The technical challenge is for the equations to develop the breakdown through the motion of the fluid itself, rather than, for example, us putting in an infinite force by hand,” OpenAI said.