Navier-Stokes and AI in mathematics

OpenAI claims to have solved a longstanding mathematical problem: the Navier-Stokes Problem. The Navier-Stokes equations describe the motion of fluids. The problem is this: given an initial state of the fluid (initial velocities of the fluid at every point and a field of forces acting on the fluid), is there always a “nice” (i.e. fluid-like)[1] solution to the equations? This problem has eluded mathematicians for almost a century. It’s one of the Clay Institute’s Millennium Prize problems carrying a $1 million bounty. And yet OpenAI claims a swarm of 10 000 agents working for 88 hours at a cost of $15 million has found a counterexample — initial conditions with non-smooth solutions. The solution is being verified at time of writing, and there is some controversy as to whether OpenAI essentially stole the work of a pair of mathematicians who had recently made advances toward a solution and used it to train or prompt the internal model. But this post isn’t about that. There has been a…