Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911945354903552 |
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| author | Sheng, Rui Wu, Peiying Yang, Jerry Zhijian Yuan, Cheng |
| author_facet | Sheng, Rui Wu, Peiying Yang, Jerry Zhijian Yuan, Cheng |
| contents | 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;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 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 $α$ and different noise levels of the measurement data ranging from 1$\%$ to 10$\%$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03801 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs Sheng, Rui Wu, Peiying Yang, Jerry Zhijian Yuan, Cheng Numerical Analysis 68T07, 65M12, 62G05 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;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 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 $α$ and different noise levels of the measurement data ranging from 1$\%$ to 10$\%$. |
| title | Solving the inverse source problem of the fractional Poisson equation by MC-fPINNs |
| topic | Numerical Analysis 68T07, 65M12, 62G05 |
| url | https://arxiv.org/abs/2407.03801 |