MC-Nonlocal-PINNs: Handling Nonlocal Operators in PINNs via Monte Carlo Sampling

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.