  In this work, we present a machine learning approach for reducing the error
when numerically solving time-dependent partial differential equations (PDE).
We use a fully convolutional LSTM network to exploit the spatiotemporal
dynamics of PDEs. The neural network serves to enhance finite-difference and
finite-volume methods (FDM/FVM) that are commonly used to solve PDEs, allowing
us to maintain guarantees on the order of convergence of our method. We train
the network on simulation data, and show that our network can reduce error by a
factor of 2 to 3 compared to the baseline algorithms. We demonstrate our method
on three PDEs that each feature qualitatively different dynamics. We look at
the linear advection equation, which propagates its initial conditions at a
constant speed, the inviscid Burgers' equation, which develops shockwaves, and
the Kuramoto-Sivashinsky (KS) equation, which is chaotic.
