Closure Discovery for Coarse-Grained Partial Differential Equations
  using Multi-Agent Reinforcement Learning
Jan-Philipp von Bassewitz Sebastian Kaltenbach Petros Koumoutsakos
Reliable predictions of critical phenomena, such as weather, wildfires and
epidemics are often founded on models described by Partial Differential
Equations (PDEs). However, simulations that capture the full range of
spatio-temporal scales in such PDEs are often prohibitively expensive.
Consequently, coarse-grained simulations that employ heuristics and empirical
closure terms are frequently utilized as an alternative. We propose a novel and
systematic approach for identifying closures in under-resolved PDEs using
Multi-Agent Reinforcement Learning (MARL). The MARL formulation incorporates
inductive bias and exploits locality by deploying a central policy represented
efficiently by Convolutional Neural Networks (CNN). We demonstrate the
capabilities and limitations of MARL through numerical solutions of the
advection equation and the Burgers' equation. Our results show accurate
predictions for in- and out-of-distribution test cases as well as a significant
speedup compared to resolving all scales.