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

arXiv: https://arxiv.org/abs/2212.12984 · HTML: https://mlsanity.com/paper/2212.12984/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 35 references that are in the map.
- First posted on arXiv 26 Dec 2022.

## Builds on (most cited first, 14 of 14)

- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [nPINNs: nonlocal Physics-Informed Neural Networks for a parametrized nonlocal universal Laplacian operator. Algorithms and Applications](https://mlsanity.com/paper/2004.04276/) (2020)
- [Normalizing field flows: Solving forward and inverse stochastic differential equations using physics-informed flow models](https://mlsanity.com/paper/2108.12956/) (2021)
- [Machine Learning and Computational Mathematics](https://mlsanity.com/paper/2009.14596/) (2020)
- [Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations](https://mlsanity.com/paper/2203.08501/) (2022)
- [Numerical methods for nonlocal and fractional models](https://mlsanity.com/paper/2002.01401/) (2020)
- [An augmented Lagrangian deep learning method for variational problems with essential boundary conditions](https://mlsanity.com/paper/2106.14348/) (2021)

## Built on by (most cited first, 1 of 1)

- [Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks](https://mlsanity.com/paper/2406.11708/) (2024)

## Abstract (arXiv)

We propose, Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which is a generalization of MC-fPINNs in citeguo2022monte, 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.