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Volume 32, Issue 4
PFNN-2: A Domain Decomposed Penalty-Free Neural Network Method for Solving Partial Differential Equations

Hailong Sheng & Chao Yang

DOI: 10.4208/cicp.OA-2022-0114

Commun. Comput. Phys., 32 (2022), pp. 980-1006.

Published online: 2022-10

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  • Abstract

A new penalty-free neural network method, PFNN-2, is presented for solving partial differential equations, which is a subsequent improvement of our previously proposed PFNN method [1]. PFNN-2 inherits all advantages of PFNN in handling the smoothness constraints and essential boundary conditions of self-adjoint problems with complex geometries, and extends the application to a broader range of non-self-adjoint time-dependent differential equations. In addition, PFNN-2 introduces an overlapping domain decomposition strategy to substantially improve the training efficiency without sacrificing accuracy. Experiments results on a series of partial differential equations are reported, which demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numerical accuracy, convergence speed, and parallel scalability.

  • Keywords

Neural network, penalty-free method, domain decomposition, initial-boundary value problem, partial differential equation.

  • AMS Subject Headings

65M55, 68T99, 76R99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CiCP-32-980, author = {Sheng , Hailong and Yang , Chao}, title = {PFNN-2: A Domain Decomposed Penalty-Free Neural Network Method for Solving Partial Differential Equations}, journal = {Communications in Computational Physics}, year = {2022}, volume = {32}, number = {4}, pages = {980--1006}, abstract = {

A new penalty-free neural network method, PFNN-2, is presented for solving partial differential equations, which is a subsequent improvement of our previously proposed PFNN method [1]. PFNN-2 inherits all advantages of PFNN in handling the smoothness constraints and essential boundary conditions of self-adjoint problems with complex geometries, and extends the application to a broader range of non-self-adjoint time-dependent differential equations. In addition, PFNN-2 introduces an overlapping domain decomposition strategy to substantially improve the training efficiency without sacrificing accuracy. Experiments results on a series of partial differential equations are reported, which demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numerical accuracy, convergence speed, and parallel scalability.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.OA-2022-0114}, url = {http://global-sci.org/intro/article_detail/cicp/21136.html} }
TY - JOUR T1 - PFNN-2: A Domain Decomposed Penalty-Free Neural Network Method for Solving Partial Differential Equations AU - Sheng , Hailong AU - Yang , Chao JO - Communications in Computational Physics VL - 4 SP - 980 EP - 1006 PY - 2022 DA - 2022/10 SN - 32 DO - http://doi.org/10.4208/cicp.OA-2022-0114 UR - https://global-sci.org/intro/article_detail/cicp/21136.html KW - Neural network, penalty-free method, domain decomposition, initial-boundary value problem, partial differential equation. AB -

A new penalty-free neural network method, PFNN-2, is presented for solving partial differential equations, which is a subsequent improvement of our previously proposed PFNN method [1]. PFNN-2 inherits all advantages of PFNN in handling the smoothness constraints and essential boundary conditions of self-adjoint problems with complex geometries, and extends the application to a broader range of non-self-adjoint time-dependent differential equations. In addition, PFNN-2 introduces an overlapping domain decomposition strategy to substantially improve the training efficiency without sacrificing accuracy. Experiments results on a series of partial differential equations are reported, which demonstrate that PFNN-2 can outperform state-of-the-art neural network methods in various aspects such as numerical accuracy, convergence speed, and parallel scalability.

Sheng , Hailong and Yang , Chao. (2022). PFNN-2: A Domain Decomposed Penalty-Free Neural Network Method for Solving Partial Differential Equations. Communications in Computational Physics. 32 (4). 980-1006. doi:10.4208/cicp.OA-2022-0114
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