Deep Domain Decomposition Methods: Helmholtz Equation (Advances in Applied Mathematics and Mechanics, 2022)
This paper proposes a deep-learning-based Robin-Robin domain decomposition method (DeepDDM) for Helmholtz equations. We first present the planewave activation-based neural network (PWNN), which is more efficient for solvingHelmholtz equations with constant coefficients and wavenumber $k$ than finite difference methods (FDM). On this basis, we use PWNN to discretize the subproblems divided by domain decomposition methods (DDM), which is the main idea of DeepDDM. This paper will investigate the number of iterations of using DeepDDM forcontinuous and discontinuous Helmholtz equations. The results demonstrate that:DeepDDM exhibits behaviors consistent with conventional robust FDM-based domaindecomposition method (FDM-DDM) under the same Robin parameters, i.e., the number of iterations by DeepDDM is almost the same as that of FDM-DDM. By choosingsuitable Robin parameters on different subdomains, the convergence rate is almostconstant with the rise of wavenumber in both continuous and discontinuous cases.The performance of DeepDDM on Helmholtz equations may provide new insights forimproving the PDE solver by deep learning.