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

TL;DR

The paper addresses the challenge of solving fractional PDEs on irregular domains where nonlocal operators complicate numerical methods. It introduces GMC-PINNs, a framework that unifies general Monte Carlo (and quasi-Monte Carlo) approximations of fractional derivatives with Physics-Informed Neural Networks to solve fPDEs across definitions such as Caputo, Grünwald-Letnikov, and Riesz derivatives. Key contributions include a Monte Carlo estimator for right-sided fractional derivatives, two quasi-Monte Carlo schemes (Sobol and Halton), and a PINN solver that uses a triple-term loss to enforce equation, initial, and boundary conditions, demonstrating superior computational efficiency and accuracy relative to existing fPINN and MC-fPINN approaches on 2D and 3D irregular domains, including fuzzy boundary scenarios. The method enables parallelizable, clutter-free computation on GPUs and broad applicability to complex geometries, with potential impact on biomedical simulations and other nonlocal systems modeled by fractional dynamics.

Abstract

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.

Figures (9)

• Figure 1: Results of the Monte Carlo method of the right-sided fractional derivatives of the function $(1-x)^2$ at different numbers of $N, K$ for different fractional orders $\alpha$. For the case $\alpha = 1.2$: (a) $K = N = 10$, $L^2~\mathrm{error} = 5.63\mathrm{e}^{-2}$. (b) $K = N = 100$, $L^2~\mathrm{error} = 9.94\mathrm{e}^{-3}$. (c) $K = N = 3000$, $L^2~\mathrm{error} = 6.1\mathrm{e}^{-5}$. For the case $\alpha = 1.6$: (d) $K = N = 10$, $L^2~\mathrm{error} = 1.11\mathrm{e}^{-1}$. (e) $K = N = 100$, $L^2~\mathrm{error} = 2.61\mathrm{e}^{-2}$. (f) $K = N = 3000$, $L^2~\mathrm{error} = 9.95\mathrm{e}^{-5}$.
• Figure 2: Results of the Quasi-Monte Carlo method (Sobol/Halton) of the left-sided and right-sided fractional derivatives of $x^2$ and $(1-x)^2$ at $N = K = 1000$ for different fractional orders $\alpha$. For left-sided fractional derivative of $x^2$: (a) $\alpha = 1.3$, $L^2~\mathrm{error} = 4.51\mathrm{e}^{-4}$ and $L^2~\mathrm{error} = 4.47\mathrm{e}^{-4}$. (b) $\alpha = 1.8$, $L^2~\mathrm{error} = 5.07\mathrm{e}^{-4}$ and $L^2~\mathrm{error} = 4.91\mathrm{e}^{-4}$. For right-sided fractional derivative of $(1-x)^2$: (c) $\alpha = 1.3$, $L^2~\mathrm{error} = 4.49\mathrm{e}^{-4}$ and $L^2~\mathrm{error} = 4.18\mathrm{e}^{-4}$. (d)$\alpha = 1.8$, $L^2~\mathrm{error} = 4.96\mathrm{e}^{-4}$ and $L^2~\mathrm{error} = 4.88\mathrm{e}^{-4}$.
• Figure 3: Schematic of the General Monte Carlo PINN method.
• Figure 4: The generating point distribution of the reduced set $\{Y_i\}$. Using the Monte Carlo method to generate a total of 100 approximation points reduces to 21, 22 respectively, for (a) $\alpha=0.5$; (b) $\alpha=1.5$, after classification.
• Figure 5: Point-wise error distribution of Example 4.1 on $\Omega_2$ under different methods. (a) fPINN method, $L^2~\mathrm{error} = 9.43\mathrm{e}^{-2}$. (b) Monte Carlo method, $L^2~\mathrm{error} = 4.72\mathrm{e}^{-2}$. (c) Sobol Monte Carlo method, $L^2~\mathrm{error} = 4.19\mathrm{e}^{-2}$. (d) Halton Monte Carlo method, $L^2~\mathrm{error} = 4.51\mathrm{e}^{-2}$. Here we set $N=16$ and the network structure to 1 layer with 200 neurons.