Navier-Stokes equations

By mathematically proving how individual molecules create the complex motion of fluids, three mathematicians have illuminated why time can’t flow in reverse.

Rose Yu has a plan for how to make AI better, faster and smarter — and it’s already yielding results.

In math and computer science, researchers have long understood that some questions are fundamentally unanswerable. Now physicists are exploring how even ordinary physical systems put hard limits on what we can predict, even in principle.

For more than 250 years, mathematicians have wondered if the Euler equations might sometimes fail to describe a fluid’s flow. A new computer-assisted proof marks a major breakthrough in that quest.

The famed Navier-Stokes equations can lead to cases where more than one result is possible, but only in an extremely narrow set of situations.

For centuries, mathematicians have tried to prove that Euler’s fluid equations can produce nonsensical answers. A new approach to machine learning has researchers betting that “blowup” is near.

Today’s powerful but little-understood artificial intelligence breakthroughs echo past examples of unexpected scientific progress.

A new proof establishes the boundary at which a shape becomes so corrugated, it can be crushed.

Two new approaches allow deep neural networks to solve entire families of partial differential equations, making it easier to model complicated systems and to do so orders of magnitude faster.