MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling
This provides a stable method for researchers and engineers dealing with high-dimensional nonlocal problems, but it is incremental as it generalizes an existing approach.
The paper tackles solving high-dimensional nonlocal models like integral equations and nonlocal PDEs by proposing MC-Nonlocal-PINNs, which uses Monte Carlo sampling to handle nonlocal operators, resulting in a stable approach demonstrated on various test problems.
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.