TY - JOUR
T1 - High Order Deep Domain Decomposition Method for Solving High Frequency Interface Problems
AU - Chang , Zhipeng
AU - Li , Ke
AU - Zou , Xiufen
AU - Xiang , Xueshuang
JO - Advances in Applied Mathematics and Mechanics
VL - 6
SP - 1602
EP - 1630
PY - 2023
DA - 2023/10
SN - 15
DO - http://doi.org/10.4208/aamm.OA-2022-0006
UR - https://global-sci.org/intro/article_detail/aamm/22053.html
KW - Deep neural network, high order methods, high-frequency interface problems, domain decomposition method.
AB - <p style="text-align: justify;">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 $(p&gt;1)$ could capture the high-frequency information very well. When compared to the deep domain decomposition method (DeepDDM), HOrderDeepDDMs $(p &gt;1)$ 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.</p>