  This paper proposes a deep-learning-based domain decomposition method
(DeepDDM), which leverages deep neural networks (DNN) to discretize the
subproblems divided by domain decomposition methods (DDM) for solving partial
differential equations (PDE). Using DNN to solve PDE is a physics-informed
learning problem with the objective involving two terms, domain term and
boundary term, which respectively make the desired solution satisfy the PDE and
corresponding boundary conditions. DeepDDM will exchange the subproblem
information across the interface in DDM by adjusting the boundary term for
solving each subproblem by DNN. Benefiting from the simple implementation and
mesh-free strategy of using DNN for PDE, DeepDDM will simplify the
implementation of DDM and make DDM more flexible for complex PDE, e.g., those
with complex interfaces in the computational domain. This paper will firstly
investigate the performance of using DeepDDM for elliptic problems, including a
model problem and an interface problem. The numerical examples demonstrate that
DeepDDM exhibits behaviors consistent with conventional DDM: the number of
iterations by DeepDDM is independent of network architecture and decreases with
increasing overlapping size. The performance of DeepDDM on elliptic problems
will encourage us to further investigate its performance for other kinds of PDE
and may provide new insights for improving the PDE solver by deep learning.
