# GMC-PINNs: A new general Monte Carlo PINNs method for solving fractional partial differential equations on irregular domains

arXiv: https://arxiv.org/abs/2405.00217 · HTML: https://mlsanity.com/paper/2405.00217/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 37 references that are in the map.
- First posted on arXiv 30 Apr 2024.

## Builds on (most cited first, 2 of 2)

- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [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)

## Built on by (most cited first, 6 of 6)

- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Tackling the Curse of Dimensionality in Fractional and Tempered Fractional PDEs with Physics-Informed Neural Networks](https://mlsanity.com/paper/2406.11708/) (2024)
- [Solving Time-Fractional Partial Integro-Differential Equations Using Tensor Neural Network](https://mlsanity.com/paper/2504.01440/) (2025)
- [MINPO: Memory-Informed Neural Pseudo-Operator to Resolve Nonlocal Spatiotemporal Dynamics](https://mlsanity.com/paper/2512.17273/) (2025)
- [Revising the Structure of Recurrent Neural Networks to Eliminate Numerical Derivatives in Forming Physics Informed Loss Terms with Respect to Time](https://mlsanity.com/paper/2409.10388/) (2024)
- [fTNN: a tensor neural network for fractional PDEs](https://mlsanity.com/paper/2606.27140/) (2026)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) have been widely used for solving partial differential equations (PDEs) of different types, including fractional PDEs (fPDES) [29]. Herein, we propose a new general (quasi) Monte Carlo PINN for solving fPDEs on irregular domains. Specifically, instead of approximating fractional derivatives by Monte Carlo approximations of integrals as was done previously in [31], we use a more general Monte Carlo approximation method to solve different fPDEs, which is valid for fractional differentiation under any definition. Moreover, based on the ensemble probability density function, the generated nodes are all located in denser regions near the target point where we perform the differentiation. This has an unexpected connection with known finite difference methods on non-equidistant or nested grids, and hence our method inherits their advantages. At the same time, the generated nodes exhibit a block-like dense distribution, leading to a good computational efficiency of this approach. We present the framework for using this algorithm and apply it to several examples. Our results demonstrate the effectiveness of GMC-PINNs in dealing with irregular domain problems and show a higher computational efficiency compared to the original fPINN method. We also include comparisons with the Monte Carlo fPINN [31]. Finally, we use examples to demonstrate the effectiveness of the method in dealing with fuzzy boundary location problems, and then use the method to solve the coupled 3D fractional Bloch-Torrey equation defined in the ventricular domain of the human brain, and compare the results with classical numerical methods.