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Volume 16, Issue 3
MC-Nonlocal-PINNs: Handling Nonlocal Operators in PINNs via Monte Carlo Sampling

Xiaodong Feng, Yue Qian & Wanfang Shen

DOI: 10.4208/nmtma.OA-2022-0201

Numer. Math. Theor. Meth. Appl., 16 (2023), pp. 769-791.

Published online: 2023-08

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  • Abstract

We propose Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which are a generalization of MC-fPINNs in L. Guo et al. (Comput. Methods Appl. Mech. Eng. 400 (2022), 115523) for solving general nonlocal models such as integral equations and nonlocal PDEs. Similar to MC-fPINNs, our MC-Nonlocal-PINNs handle nonlocal operators in a Monte Carlo way, resulting in a very stable approach for high dimensional problems. We present a variety of test problems, including high dimensional Volterra type integral equations, hypersingular integral equations and nonlocal PDEs, to demonstrate the effectiveness of our approach.

  • Keywords

Nonlocal models, PINNs, Monte Carlo sampling, deep neural networks.

  • AMS Subject Headings

65C05, 65D30, 65R20

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{NMTMA-16-769, author = {Feng , XiaodongQian , Yue and Shen , Wanfang}, title = {MC-Nonlocal-PINNs: Handling Nonlocal Operators in PINNs via Monte Carlo Sampling}, journal = {Numerical Mathematics: Theory, Methods and Applications}, year = {2023}, volume = {16}, number = {3}, pages = {769--791}, abstract = {

We propose Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which are a generalization of MC-fPINNs in L. Guo et al. (Comput. Methods Appl. Mech. Eng. 400 (2022), 115523) for solving general nonlocal models such as integral equations and nonlocal PDEs. Similar to MC-fPINNs, our MC-Nonlocal-PINNs handle nonlocal operators in a Monte Carlo way, resulting in a very stable approach for high dimensional problems. We present a variety of test problems, including high dimensional Volterra type integral equations, hypersingular integral equations and nonlocal PDEs, to demonstrate the effectiveness of our approach.

}, issn = {2079-7338}, doi = {https://doi.org/10.4208/nmtma.OA-2022-0201}, url = {http://global-sci.org/intro/article_detail/nmtma/21966.html} }
TY - JOUR T1 - MC-Nonlocal-PINNs: Handling Nonlocal Operators in PINNs via Monte Carlo Sampling AU - Feng , Xiaodong AU - Qian , Yue AU - Shen , Wanfang JO - Numerical Mathematics: Theory, Methods and Applications VL - 3 SP - 769 EP - 791 PY - 2023 DA - 2023/08 SN - 16 DO - http://doi.org/10.4208/nmtma.OA-2022-0201 UR - https://global-sci.org/intro/article_detail/nmtma/21966.html KW - Nonlocal models, PINNs, Monte Carlo sampling, deep neural networks. AB -

We propose Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which are a generalization of MC-fPINNs in L. Guo et al. (Comput. Methods Appl. Mech. Eng. 400 (2022), 115523) for solving general nonlocal models such as integral equations and nonlocal PDEs. Similar to MC-fPINNs, our MC-Nonlocal-PINNs handle nonlocal operators in a Monte Carlo way, resulting in a very stable approach for high dimensional problems. We present a variety of test problems, including high dimensional Volterra type integral equations, hypersingular integral equations and nonlocal PDEs, to demonstrate the effectiveness of our approach.

Feng , XiaodongQian , Yue and Shen , Wanfang. (2023). MC-Nonlocal-PINNs: Handling Nonlocal Operators in PINNs via Monte Carlo Sampling. Numerical Mathematics: Theory, Methods and Applications. 16 (3). 769-791. doi:10.4208/nmtma.OA-2022-0201
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