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.

提出了一种用于解决不规则域上分数阶偏微分方程（fPDEs）的新型广义蒙特卡洛物理信息神经网络（GMC-PINN）方法，并通过多个示例表明其有效性和计算效率高于原始的 fPINN 方法。