A $1 million prize awaits anyone who can show where the math of fluid flow breaks down. With specially trained AI systems, researchers have found a slew of new candidates in simpler versions of the problem.

Nearly 200 years ago, the physicists Claude-Louis Navier and George Gabriel Stokes put the finishing touches on a set of equations that describe how fluids swirl. And for nearly 200 years, the Navier-Stokes equations have served as an unimpeachable theory of how fluids in the real world behave — from ocean currents threading their way between the continents to air wrapping around an aircraft’s wings.

Nevertheless, many mathematicians suspect that glitches hide deep within the equations. They have a hunch that in certain situations, the theory fails. In these cases, the equations will predict a fluid moving in some unphysical, incomprehensible way — spinning into an impossibly fast vortex, for instance, or instantly reversing its flow. Some quantity in the equations will grow infinitely large, or “blow up,” as mathematicians put it.

Despite immense effort, no one has been able to come up with a situation where the Navier-Stokes equations falter. Doing so — or, alternatively, proving that the equations never blow up — would come with a $1 million reward. And so, as a prelude to solving the Navier-Stokes problem, mathematicians have searched for blowups (also called singularities) in an assortment of simplified fluid equations, such as those that operate in only one dimension.

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