  Adaptive mesh refinement (AMR) is necessary for efficient finite element
simulations of complex physical phenomenon, as it allocates limited
computational budget based on the need for higher or lower resolution, which
varies over space and time. We present a novel formulation of AMR as a
fully-cooperative Markov game, in which each element is an independent agent
who makes refinement and de-refinement choices based on local information. We
design a novel deep multi-agent reinforcement learning (MARL) algorithm called
Value Decomposition Graph Network (VDGN), which solves the two core challenges
that AMR poses for MARL: posthumous credit assignment due to agent creation and
deletion, and unstructured observations due to the diversity of mesh
geometries. For the first time, we show that MARL enables anticipatory
refinement of regions that will encounter complex features at future times,
thereby unlocking entirely new regions of the error-cost objective landscape
that are inaccessible by traditional methods based on local error estimators.
Comprehensive experiments show that VDGN policies significantly outperform
error threshold-based policies in global error and cost metrics. We show that
learned policies generalize to test problems with physical features, mesh
geometries, and longer simulation times that were not seen in training. We also
extend VDGN with multi-objective optimization capabilities to find the Pareto
front of the tradeoff between cost and error.
