  We introduce Backpropagation Through Time and Space (BPTTS), a method for
training a recurrent spatio-temporal neural network, that is used in a
homogeneous multi-agent reinforcement learning (MARL) setting to learn
numerical methods for hyperbolic conservation laws. We treat the numerical
schemes underlying partial differential equations (PDEs) as a Partially
Observable Markov Game (POMG) in Reinforcement Learning (RL). Similar to
numerical solvers, our agent acts at each discrete location of a computational
space for efficient and generalizable learning. To learn higher-order spatial
methods by acting on local states, the agent must discern how its actions at a
given spatiotemporal location affect the future evolution of the state. The
manifestation of this non-stationarity is addressed by BPTTS, which allows for
the flow of gradients across both space and time. The learned numerical
policies are comparable to the SOTA numerics in two settings, the Burgers'
Equation and the Euler Equations, and generalize well to other simulation
set-ups.
