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

TL;DR

This paper introduces MC-fPINNs, a neural network-based method that effectively solves the inverse source problem of the fractional Poisson equation, demonstrating high accuracy and robustness in high-dimensional and noisy data scenarios.

Contribution

The paper develops MC-fPINNs, combining neural networks with Monte Carlo sampling, for solving inverse fractional Poisson problems with error analysis and parameter selection guidelines.

Findings

High accuracy in 10D problems

Robustness against 10% noise levels

Effective in various fractional orders

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;\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$\%$.