Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs
<|reference_start|>Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs: 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$\%$.<|reference_end|>
@article{sheng2024solving,
    title={Solving the inverse source problem of the fractional Poisson equation by
  MC-fPINNs},
    author={Rui Sheng, Peiying Wu, Jerry Zhijian Yang, Cheng Yuan},
    journal={arXiv preprint arXiv:2407.03801},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.03801},
    primaryClass={math.NA cs.NA}
}