@Article{AAMM-15-6,
author = {Zhipeng, Chang and Li, Ke and Zou, Xiufen and Xueshuang, Xiang},
title = {High Order Deep Domain Decomposition Method for Solving High Frequency Interface Problems},
journal = {Advances in Applied Mathematics and Mechanics},
year = {2023},
volume = {15},
number = {6},
pages = {1602--1630},
abstract = {<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>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.OA-2022-0006},
url = {https://global-sci.com/article/72885/high-order-deep-domain-decomposition-method-for-solving-high-frequency-interface-problems}
}