Data-driven discretization of PDEs

One of the most generic problems in theoretical physics in general, and in Fluid Mechanics in particular, is that of coarse graining: how to represent the behavior of a physical theory at long wavelengths and slow frequencies by "integrating out" degrees of freedom which change rapidly in time and space. This is crucial for allowing an efficient computation of the field equations, as otherwise the number of grid points required to simulate a given system becomes unmanagable. Here we introduce data-driven discretization, a method for learning the 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 (PDEs). We demonstrate that this approach obtains remarkable accuracy which allows to integrate equations in time and to extract emerging scaling relations in one and two spatial dimensions. Possible applications and generalizations are discussed.