High Order Deep Neural Network for Solving High Frequency Partial Differential Equations
作者机构：Qian Xuesen Laboratory of Space TechnologyChina Academy of Space TechnologyBeijing 100875P.R.China School of Mathematics and StatisticsWuhan UniversityWuhan 430072P.R.China Information Engineering UniversityZhengzhou 450001P.R.China
出 版 物：《Communications in Computational Physics》 (计算物理通讯（英文）)
年 卷 期：2022年第31卷第2期
页 面：370-397页
核心收录：
学科分类：07[理学] 0701[理学-数学] 0702[理学-物理学] 070101[理学-基础数学]
基 金：supported partly by National Key R&D Program of China with grants 2019YFA0709600,2019YFA0709602 National Natural Science Foundation of China with grants 11831016,12101609 the Innovation Foundation of Qian Xuesen Laboratory of Space Technology
主 题：Deep neural network high order methods high frequency PDEs
摘 要：This paper proposes a high order deep neural network(HOrderDNN)for solving high frequency partial differential equations(PDEs),which incorporates the idea of“high orderfrom finite element methods(FEMs)into commonly-used deep neural networks(DNNs)to obtain greater approximation *** main idea of HOrderDNN is introducing a nonlinear transformation layer between the input layer and the first hidden layer to form a high order polynomial space with the degree not exceeding p,followed by a normal *** order p can be guided by the regularity of solutions of *** performance of HOrderDNNis evaluated on high frequency function fitting problems and high frequency Poisson and Helmholtz *** results demonstrate that:HOrderDNNs(p1)can efficiently capture the high frequency information in target functions;and when compared to physics-informed neural network(PINN),HOrderDNNs(p1)converge faster and achieve much smaller relative errors with same number of trainable *** particular,when solving the high frequency Helmholtz equation in Section 3.5,the relative error of PINN stays around 1 with its depth and width increase,while the relative error can be reduced to around 0.02 as p increases(see Table 5).
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