Learning data-driven discretizations for partial differential equations

In many physical systems, the governing equations are known with high confidence, but direct numerical solution is prohibitively expensive. Often this situation is alleviated by writing effective equations to approximate dynamics below the grid scale. This process is often impossible to perform analytically and is often ad hoc. Here we propose data-driven discretization, a method that uses machine learning to systematically derive discretizations for continuous physical systems. On a series of model problems, data-driven discretization gives accurate solutions with a dramatic drop in required resolution.