Year: 2025
Author: Rui Sheng, Peiying Wu, Jerry Zhijian Yang, Cheng Yuan
East Asian Journal on Applied Mathematics, Vol. 15 (2025), Iss. 3 : pp. 565–590
Abstract
In this paper, we effectively solve the inverse source problem of the fractional Poisson equation using a Monte Carlo sampling-based PINN method (MC-fPINN). 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 networks, we use the Monte Carlo sampling method and define a new loss function combining the measurement data and 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. Numerical examples demonstrate the great accuracy and robustness of the method in solving high-dimensional problems up to 10D, with various fractional orders and noise levels of the measurement data ranging from 1% to 10%.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/eajam.2024-072.150824
East Asian Journal on Applied Mathematics, Vol. 15 (2025), Iss. 3 : pp. 565–590
Published online: 2025-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 26
Keywords: Fractional Poisson equation MC-fPINN error analysis inverse source problem.