OpenAI claims to have solved the Navier–Stokes Millennium Prize Problem. Read the announcement.
The Navier–Stokes equations are a set of equations that describe how fluids behave. This is key for aerodynamics (airplanes), hydrodynamics (ships) and everything else that deals with flowing gases or liquids (like blood flow).
Unfortunately these equations are non-linear partial differential equations (PDEs). Nonlinear means that the effect of a change in the input can depend on the current state of the system. For example if I double the input of a linear function I get double the output. For a nonlinear function this does not generally hold.
A differential equation is an equation that relates the properties of a system to their rates of change. Partial differential equations describe changes with respect to multiple variables, such as space and time. Many processes in nature are described by these types of equations (like the weather models I posted about the other day).
Because of the non-linearity and the complexity of realistic flow conditions, it is usually not possible to calculate an exact analytical solution to these PDEs. Instead, they have to be solved approximately through simulations, often with high performance computers.
A simulation divides space into small cells and time into small steps. For each cell we start with values for the physical properties (for example temperature and pressure and the speed and direction of the flow of a fluid). Then we use the properties in neighbouring cells to estimate spatial changes and apply the PDEs to estimate how each cell's properties evolve in the next time step. We update the current properties and move to the next time step.
Each step introduces numerical errors that can add up to a larger error after many steps. For realistic flows, even with the most powerful simulations you get an approximation rather than an exact solution. In weather forecasts, uncertainties in the initial conditions and approximations in the physics also matter, and the chaotic nature of the atmosphere amplifies small errors. That's why a weather forecast that is 10 days out is less accurate than the forecast for the next hour. Simulations alone cannot establish whether smooth solutions always remain smooth, which is the subject of the Millennium Prize Problem.
A Millennium Prize Problem is a problem that is supposed to be one of the biggest mathematical problems of our time. An accepted solution can earn a prize of 1 million USD. In this case, the problem is to find out whether the three-dimensional incompressible Navier–Stokes equations guarantee that a smooth flow stays smooth forever or if there is a case where the smoothness breaks down.
OpenAI says they found an example where this smoothness breaks down under a smooth external force: The velocity of the fluid grows without bound as it approaches a finite time. They claim to have found an analytical proof. They used roughly 10,000 concurrent agents of a new internal model to solve this.
The agents arrived at their resolution on Saturday, September 5, about 88 hours after the first agents were launched. Lean formalization and verification took an additional 17 hours via GPT‑6 Astra.
— OpenAI
If this proof is validated, it gives us a glimpse of what impact AI might have in the physical world. Right now we see most of the changes in the digital world, specifically in software engineering. But there seems to be a path where AI models can also lead to enormous changes in science and engineering, through new insights that eventually inform better physical models and engineering methods. Truly exciting times!
Header image: Unsplash.