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 - <p style="text-align: justify;">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.</p>