In this paper, we effectively solve the inverse source problem of the fractional Poisson equation using MC-fPINNs. We construct two neural networks $ u_{NN}(x;\theta )$ and $f_{NN}(x;\psi)$ to approximate the solution $u^{*}(x)$ and the forcing term $f^{*}(x)$ of the fractional Poisson equation. To optimize these two neural networks, we use the Monte Carlo sampling method mentioned in MC-fPINNs and define a new loss function combining measurement data and the underlying physical model. Meanwhile, we present a comprehensive error analysis for this method, along with a prior rule to select the appropriate parameters of neural networks. Several numerical examples are given to demonstrate the great precision and robustness of this method in solving high-dimensional problems up to 10D, with various fractional order $\alpha$ and different noise levels of the measurement data ranging from 1$\%$ to 10$\%$.

翻译：本文利用MC-fPINNs有效求解了分数阶泊松方程的反源问题。我们构建了两个神经网络 $ u_{NN}(x;\theta )$ 和 $f_{NN}(x;\psi)$，分别用于逼近分数阶泊松方程的解 $u^{*}(x)$ 和源项 $f^{*}(x)$。为优化这两个神经网络，我们采用了MC-fPINNs中提及的蒙特卡洛采样方法，并定义了一个结合测量数据与底层物理模型的新损失函数。同时，我们对此方法进行了完整的误差分析，并提出了用于选取神经网络合适参数的先验准则。通过多个数值算例验证了该方法在求解高达10维的问题时具有高精度和强鲁棒性，这些算例涵盖了不同的分数阶 $\alpha$ 以及测量数据从1$\%$到10$\%$的不同噪声水平。