Data-driven discretization: a method for systematic coarse graining of partial differential equations

Many problems in theoretical physics are centered on representing the behavior of a physical theory at long wave lengths and slow frequencies by integrating out degrees of freedom which change rapidly in time and space. This is typically done by writing effective long-wavelength governing equations which take into account the small scale physics, a procedure which is very difficult to perform analytically, and sometimes even phenomenologically. Here we introduce \emph{data driven discretization}, a method for automatically learning effective long-wavelength dynamics from actual solutions to the known underlying equations. We use a neural network to learn a discretization for the true spatial derivatives of partial differential equations. We demonstrate that this approach is remarkably accurate, allowing us to integrate in time a collection of nonlinear equations in one spatial dimension at resolutions 4-8x coarser than is possible with standard finite difference methods.