@Article{CiCP-31-2,
author = {Zhipeng, Chang and Li, Ke and Zou, Xiufen and Xueshuang, Xiang},
title = {High Order Deep Neural Network for Solving High Frequency Partial Differential Equations},
journal = {Communications in Computational Physics},
year = {2022},
volume = {31},
number = {2},
pages = {370--397},
abstract = {<p style="text-align: justify;">This paper proposes a high order deep neural network (HOrderDNN) for
solving high frequency partial differential equations (PDEs), which incorporates the
idea of &quot;high order&quot; from finite element methods (FEMs) into commonly-used deep
neural networks (DNNs) to obtain greater approximation ability. The 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 DNN. The order $p$ can be guided by the regularity
of solutions of PDEs. The performance of HOrderDNN is evaluated on high frequency
function fitting problems and high frequency Poisson and Helmholtz equations. The
results demonstrate that: HOrderDNNs($p &gt; 1$) can efficiently capture the high frequency information in target functions; and when compared to physics-informed neural network (PINN), HOrderDNNs($p &gt; 1$) converge faster and achieve much smaller
relative errors with same number of trainable parameters. In 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).</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2021-0092},
url = {https://global-sci.com/article/79516/high-order-deep-neural-network-for-solving-high-frequency-partial-differential-equations}
}