“for nearly 200 years, the Navier-Stokes equations have served as an unimpeachable theory of how fluids in the real world behave”
“mathematicians suspect that glitches hide deep within the equations. They have a hunch that in certain situations, the theory fails.”
“mathematicians have searched for blowups (also called singularities) in an assortment of simplified fluid equations, such as those that operate in only one dimension.”
“The great mystery is whether every solution — every single possible fluid history that satisfies the Navier-Stokes equations — makes sense everywhere and always.”
“they used a neural network as a new way to search indiscriminately for singularity candidates of any kind. A neural network is, in general, a function defined by a vast array of numbers. These numbers get carefully adjusted through a highly efficient “training” process of guessing, checking, and refining until the function can perform some desired task.”
“Buckmaster and his team turned to what’s known as a physics-informed neural network, or PINN. Unlike an image-classifying neural network, a PINN doesn’t learn by studying external data. Instead, it takes a partial differential equation — an equation that describes how a system changes over time — and adjusts itself until it can represent a function that solves that equation.”