MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling

TL;DR

This paper introduces MC-Nonlocal-PINNs, a Monte Carlo-based neural network method for solving complex nonlocal models like integral equations and PDEs, demonstrating stability and effectiveness in high-dimensional problems.

Contribution

It extends MC-fPINNs to handle general nonlocal operators, providing a stable and effective approach for high-dimensional nonlocal models.

Findings

Successfully solves high-dimensional Volterra integral equations

Handles hypersingular integral equations effectively

Demonstrates stability and accuracy in nonlocal PDEs

Abstract

We propose, Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which is a generalization of MC-fPINNs in \cite{guo2022monte}, for solving general nonlocal models such as integral equations and nonlocal PDEs. Similar as in MC-fPINNs, our MC-Nonlocal-PINNs handle the 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.