High Order Deep Domain Decomposition Method for Solving High Frequency Interface Problems (Advances in Applied Mathematics and Mechanics, 2023)
This paper proposes a high order deep domain decomposition method(HOrderDeepDDM) for solving high-frequency interface problems, which combineshigh order deep neural network (HOrderDNN) with domain decomposition method(DDM). The main idea of HOrderDeepDDM is to divide the computational domaininto some sub-domains by DDM, and apply HOrderDNNs to solve the high-frequencyproblem on each sub-domain. Besides, we consider an adaptive learning rate annealing method to balance the errors inside the sub-domains, on the interface and theboundary during the optimization process. The performance of HOrderDeepDDM isevaluated on high-frequency elliptic and Helmholtz interface problems. The results indicate that: HOrderDeepDDM inherits the ability of DeepDDM to handle discontinuous interface problems and the power of HOrderDNN to approximate high-frequencyproblems. In detail, HOrderDeepDDMs $(p>1)$ could capture the high-frequency information very well. When compared to the deep domain decomposition method (DeepDDM), HOrderDeepDDMs $(p >1)$ converge faster and achieve much smaller relativeerrors with the same number of trainable parameters. For example, when solving thehigh-frequency interface elliptic problems in Section 3.3.1, the minimum relative errors obtained by HOrderDeepDDMs $(p =9)$ are one order of magnitude smaller thanthat obtained by DeepDDMs when the number of the parameters keeps the same, asshown in Fig. 4.