What Is Navier-Stokes, the ‘Millennium’ Math Problem OpenAI Claims to Have Solved?

On Tuesday, OpenAI announced that its latest A.I. technology, which has not yet been released to the public, solved an enormously complex math problem called “the Navier-Stokes existence and smoothness problem.” Mathematicians sometimes call it “Navier-Stokes” for short.

Here is a brief guide to the problem.

Why is Navier-Stokes important?

It is one of seven “Millennium Problems” selected in the year 2000 by a nonprofit called the Clay Mathematics Institute as a way of advancing mathematics in the new millennium.

The institute, founded by a businessman named Landon T. Clay, offered a $1 million prize to the first mathematician who could solve each of the problems. Before OpenAI’s announcement, only one of the seven had been solved.

What does the problem explore?

It looks at a series of mathematical equations that describe the movement of liquids, like water, and gases, like air.

These equations are often used to predict the weather. They can also be used to model the behavior of ocean currents or the climate as a whole. But mathematicians have struggled to understand parts of these equations.

The Navier-Stokes problem essentially asks if these equations become unreliable in certain situations; if there are cases where they can no longer be solved. This would imply, theoretically, that the laws of physics would break down under certain extreme conditions, that, for example, water could be made to spontaneously explode.

“Is there some initial fluid configuration where water splashes around and, at some point, all the energy implodes and sends all the water particles off into infinite velocity?” said Terence Tao, a professor at the University of California, Los Angeles, describing the implications of the result. “Could you set some fluid in motion, watch it flow, and then it blows up?”

Could that ever happen?

Not that we know of. And Dr. Tao adds that simulations of water flow have not shown that it could happen, either. But in its solution to the Navier-Stokes problem, OpenAI seems to have shown that it could happen in theory.