High Order Deep Domain Decomposition Method for Solving High Frequency Interface Problems

Abstract

This paper proposes a high order deep domain decomposition method (HOrderDeepDDM) for solving high-frequency interface problems, which combines high order deep neural network (HOrderDNN) with domain decomposition method (DDM). The main idea of HOrderDeepDDM is to divide the computational domain into some sub-domains by DDM, and apply HOrderDNNs to solve the high-frequency problem 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 the boundary during the optimization process. The performance of HOrderDeepDDM is evaluated 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-frequency problems. 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 relative errors with the same number of trainable parameters. For example, when solving the high-frequency interface elliptic problems in Section 3.3.1, the minimum relative errors obtained by HOrderDeepDDMs $(p =9)$ are one order of magnitude smaller than that obtained by DeepDDMs when the number of the parameters keeps the same, as shown in Fig. 4.