arrow
Volume 32, Issue 2
MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs

Lulu Zhang, Tao Luo, Yaoyu Zhang, Weinan E, Zhi-Qin John Xu & Zheng Ma

Commun. Comput. Phys., 32 (2022), pp. 299-335.

Published online: 2022-08

Export citation
  • 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.

  • Keywords

Deep neural network, radiative transfer equation, Green’s method, neural operator.

  • AMS Subject Headings

35C15, 35J05, 35Q20, 35Q49, 45K05

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CiCP-32-299, author = {Lulu and Zhang and and 24267 and and Lulu Zhang and Tao and Luo and and 24268 and and Tao Luo and Yaoyu and Zhang and and 24269 and and Yaoyu Zhang and Weinan and E and and 24270 and and Weinan E and Zhi-Qin and John Xu and and 24271 and and Zhi-Qin John Xu and Zheng and Ma and and 24272 and and Zheng Ma}, title = {MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs}, journal = {Communications in Computational Physics}, year = {2022}, volume = {32}, number = {2}, pages = {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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2021-0257}, url = {http://global-sci.org/intro/article_detail/cicp/20860.html} }
TY - JOUR T1 - MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs AU - Zhang , Lulu AU - Luo , Tao AU - Zhang , Yaoyu AU - E , Weinan AU - John Xu , Zhi-Qin AU - Ma , Zheng JO - Communications in Computational Physics VL - 2 SP - 299 EP - 335 PY - 2022 DA - 2022/08 SN - 32 DO - http://doi.org/10.4208/cicp.OA-2021-0257 UR - https://global-sci.org/intro/article_detail/cicp/20860.html KW - Deep neural network, radiative transfer equation, Green’s method, neural operator. AB -

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.

Lulu Zhang, Tao Luo, Yaoyu Zhang, Weinan E, Zhi-Qin John Xu & Zheng Ma. (2022). MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs. Communications in Computational Physics. 32 (2). 299-335. doi:10.4208/cicp.OA-2021-0257
Copy to clipboard
The citation has been copied to your clipboard