Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs

TL;DR

This work addresses the inverse source problem for the space fractional Poisson equation $(-\Delta)^{\alpha/2}u=f$ by introducing MC-fPINNs, a framework that uses two neural networks to simultaneously approximate the solution $u^*$ and the forcing term $f^*$. The method leverages Monte Carlo sampling to evaluate the nonlocal fractional Laplacian, builds a loss that couples PDE residuals with measurement data, and provides a rigorous error analysis including approximation and statistical error bounds, plus guidelines for neural network design. Numerical experiments demonstrate high accuracy and robustness in both low and high dimensions (up to $d=10$) and across fractional orders $\alpha$ with noise levels from $1\%$ to $10\%$, illustrating effective scalability to challenging inverse problems. The approach offers a scalable, theoretically grounded tool for recovering sources in nonlocal fractional PDEs, with potential application to physics-informed modeling in high-dimensional settings.

Abstract

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;θ)$ and $f_{NN}(x;ψ)$ 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 $α$ and different noise levels of the measurement data ranging from 1$\%$ to 10$\%$.

Figures (6)

• Figure 1: The reconstructions of $f$ (top) and the corresponding point-wise absolute error $|\hat{f}-f^{*}|$ (bottom) when $\alpha=0.5$
• Figure 2: The reconstructions of $f$ (top) and the corresponding point-wise absolute error $|\hat{f}-f^{*}|$ (bottom) when $\alpha=1.2$
• Figure 3: The reconstructions of $f$ (top) and the corresponding point-wise absolute error $|\hat{f}-f^{*}|$ (bottom) when $\alpha=1.5$
• Figure 4: The reconstructions of $f$ (top) and the corresponding point-wise absolute error $|\hat{f}-f^{*}|$ (bottom) when $\alpha=1.8$
• Figure 5: 3D The reconstruction of $f$ and the corresponding point-wise absolute error $|\hat{f}-f^{*}|$ (bottom) at $x_3=0.5$ when $\alpha=1.5$

Theorems & Definitions (25)

• Proposition 4.1
• Lemma 4.1
• Lemma 4.2
• Theorem 4.1
• Corollary 4.1
• Corollary 4.2
• Lemma 4.3
• Proposition 4.2
• Lemma 4.4
• Definition 4.1