TY - JOUR
T1 - Solving the Inverse Source Problem of the Fractional Poisson Equation by MC-fPINNs
AU - Sheng , Rui
AU - Wu , Peiying
AU - Yang , Jerry Zhijian
AU - Yuan , Cheng
JO - East Asian Journal on Applied Mathematics
VL - 3
SP - 565
EP - 590
PY - 2025
DA - 2025/06
SN - 15
DO - http://doi.org/10.4208/eajam.2024-072.150824
UR - https://global-sci.org/intro/article_detail/eajam/24156.html
KW - Fractional Poisson equation, MC-fPINN, error analysis, inverse source problem.
AB - <p style="text-align: justify;">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%.</p>