  Finite element discretizations of problems in computational physics often
rely on adaptive mesh refinement (AMR) to preferentially resolve regions
containing important features during simulation. However, these spatial
refinement strategies are often heuristic and rely on domain-specific knowledge
or trial-and-error. We treat the process of adaptive mesh refinement as a
local, sequential decision-making problem under incomplete information,
formulating AMR as a partially observable Markov decision process. Using a deep
reinforcement learning approach, we train policy networks for AMR strategy
directly from numerical simulation. The training process does not require an
exact solution or a high-fidelity ground truth to the partial differential
equation at hand, nor does it require a pre-computed training dataset. The
local nature of our reinforcement learning formulation allows the policy
network to be trained inexpensively on much smaller problems than those on
which they are deployed. The methodology is not specific to any particular
partial differential equation, problem dimension, or numerical discretization,
and can flexibly incorporate diverse problem physics. To that end, we apply the
approach to a diverse set of partial differential equations, using a variety of
high-order discontinuous Galerkin and hybridizable discontinuous Galerkin
finite element discretizations. We show that the resultant deep reinforcement
learning policies are competitive with common AMR heuristics, generalize well
across problem classes, and strike a favorable balance between accuracy and
cost such that they often lead to a higher accuracy per problem degree of
freedom.
