D3M: A deep domain decomposition method for partial differential equations
This provides a foundation for using variational deep learning in large-scale engineering problems, though it appears incremental as it builds on existing domain decomposition and neural network methods.
The authors tackled solving partial differential equations (PDEs) by proposing a deep domain decomposition method (D3M) based on variational principles, resulting in a framework that converges to exact solutions and is validated for accuracy and efficiency in numerical experiments.
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.