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arXiv · 2407.03801

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

Rui Sheng·Peiying Wu·Jerry Zhijian Yang·Cheng Yuan
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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$\%$.

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Rui Sheng, Peiying Wu, Jerry Zhijian Yang, Cheng Yuan. 2024-07-04. Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs. https://arxiv.org/abs/2407.03801

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