  The numerical solution of partial differential equations (PDEs) is
challenging because of the need to resolve spatiotemporal features over wide
length and timescales. Often, it is computationally intractable to resolve the
finest features in the solution. The only recourse is to use approximate
coarse-grained representations, which aim to accurately represent
long-wavelength dynamics while properly accounting for unresolved small scale
physics. Deriving such coarse grained equations is notoriously difficult, and
often \emph{ad hoc}. Here we introduce \emph{data driven discretization}, a
method for learning optimized approximations to PDEs based on actual solutions
to the known underlying equations. Our approach uses neural networks to
estimate spatial derivatives, which are optimized end-to-end to best satisfy
the equations on a low resolution grid. The resulting numerical methods are
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.
