Year: 2022
Author: Lulu Zhang, Tao Luo, Yaoyu Zhang, Weinan E, Zhi-Qin John Xu, Zheng Ma
Communications in Computational Physics, Vol. 32 (2022), Iss. 2 : pp. 299–335
Abstract
In this paper, we propose a machine learning approach via model-operator-data network (MOD-Net) for solving PDEs. A MOD-Net is driven by a model to solve PDEs based on operator representation with regularization from data. For linear PDEs, we use a DNN to parameterize the Green’s function and obtain the neural operator to approximate the solution according to the Green’s method. To train the DNN, the empirical risk consists of the mean squared loss with the least square formulation or the variational formulation of the governing equation and boundary conditions. For complicated problems, the empirical risk also includes a few labels, which are computed on coarse grid points with cheap computation cost and significantly improves the model accuracy. Intuitively, the labeled dataset works as a regularization in addition to the model constraints. The MOD-Net solves a family of PDEs rather than a specific one and is much more efficient than original neural operator because few expensive labels are required. We numerically show MOD-Net is very efficient in solving Poisson equation and one-dimensional radiative transfer equation. For nonlinear PDEs, the nonlinear MOD-Net can be similarly used as an ansatz for solving nonlinear PDEs, exemplified by solving several nonlinear PDE problems, such as the Burgers equation.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/cicp.OA-2021-0257
Communications in Computational Physics, Vol. 32 (2022), Iss. 2 : pp. 299–335
Published online: 2022-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 37
Keywords: Deep neural network radiative transfer equation Green’s method neural operator.
Author Details
-
Physics-Informed Neural Operator for Learning Partial Differential Equations
Li, Zongyi | Zheng, Hongkai | Kovachki, Nikola | Jin, David | Chen, Haoxuan | Liu, Burigede | Azizzadenesheli, Kamyar | Anandkumar, AnimaACM / IMS Journal of Data Science, Vol. 1 (2024), Iss. 3 P.1
https://doi.org/10.1145/3648506 [Citations: 25] -
DNN-HDG: A deep learning hybridized discontinuous Galerkin method for solving some elliptic problems
Baharlouei, S. | Mokhtari, R. | Mostajeran, F.Engineering Analysis with Boundary Elements, Vol. 151 (2023), Iss. P.656
https://doi.org/10.1016/j.enganabound.2023.03.039 [Citations: 4] -
Overview Frequency Principle/Spectral Bias in Deep Learning
Xu, Zhi-Qin John | Zhang, Yaoyu | Luo, TaoCommunications on Applied Mathematics and Computation, Vol. (2024), Iss.
https://doi.org/10.1007/s42967-024-00398-7 [Citations: 2] -
Bayesian Inversion with Neural Operator (BINO) for modeling subdiffusion: Forward and inverse problems
Yan, Xiong-Bin | Xu, Zhi-Qin John | Ma, ZhengJournal of Computational and Applied Mathematics, Vol. 454 (2025), Iss. P.116191
https://doi.org/10.1016/j.cam.2024.116191 [Citations: 0] -
Render unto Numerics: Orthogonal Polynomial Neural Operator for PDEs with Nonperiodic Boundary Conditions
Liu, Ziyuan | Wang, Haifeng | Zhang, Hong | Bao, Kaijun | Qian, Xu | Song, SongheSIAM Journal on Scientific Computing, Vol. 46 (2024), Iss. 4 P.C323
https://doi.org/10.1137/23M1556320 [Citations: 0] -
Capturing the diffusive behavior of the multiscale linear transport equations by Asymptotic-Preserving Convolutional DeepONets
Wu, Keke | Yan, Xiong-Bin | Jin, Shi | Ma, ZhengComputer Methods in Applied Mechanics and Engineering, Vol. 418 (2024), Iss. P.116531
https://doi.org/10.1016/j.cma.2023.116531 [Citations: 3]