  A state-of-the-art deep domain decomposition method (D3M) based on the
variational principle is proposed for partial differential equations (PDEs).
The solution of PDEs can be formulated as the solution of a constrained
optimization problem, and we design a multi-fidelity neural network framework
to solve this optimization problem. Our contribution is to develop a
systematical computational procedure for the underlying problem in parallel
with domain decomposition. Our analysis shows that the D3M approximation
solution converges to the exact solution of underlying PDEs. Our proposed
framework establishes a foundation to use variational deep learning in
large-scale engineering problems and designs. We present a general mathematical
framework of D3M, validate its accuracy and demonstrate its efficiency with
numerical experiments.
