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Artificial neural network approximations of Cauchy inverse problem for linear PDEs

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  • Li, Yixin
  • Hu, Xianliang

Abstract

A novel artificial neural network method is proposed for solving Cauchy inverse problems. Using multiple-layers network as an approximation we present a non-mesh discretization to solve the problems. The existence and convergence are shown to establish the well-posedness of neural network approximations for the Cauchy inverse problems. Numerical results on 2D to 8D cases show that compared to finite element method, the neural network approach easier extends to high dimensional case. The stability and accuracy of the proposed network approach are investigated by the experiments with noisy boundary and irregular computational domain. Our studies conclude that the neural network method alleviates the influence of noise and it is observed that networks with wider and deeper hidden layers could lead to better approximation.

Suggested Citation

  • Li, Yixin & Hu, Xianliang, 2022. "Artificial neural network approximations of Cauchy inverse problem for linear PDEs," Applied Mathematics and Computation, Elsevier, vol. 414(C).
    • Handle: RePEc:eee:apmaco:v:414:y:2022:i:c:s0096300321007621
    DOI: 10.1016/j.amc.2021.126678
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    References listed on IDEAS

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    1. S. Li, Y. & Wei, T., 2018. "An inverse time-dependent source problem for a time–space fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 257-271.
    2. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
    3. Пигнастый, Олег & Koжевников, Георгий, 2019. "Распределенная Динамическая Pde-Модель Программного Управления Загрузкой Технологического Оборудования Производственной Линии [Distributed dynamic PDE-model of a program control by utilization of t," MPRA Paper 93278, University Library of Munich, Germany, revised 02 Feb 2019.
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    Cited by:

    1. Yule Lin & Xiaoyi Yan & Jiguang Sun & Juan Liu, 2024. "Deep Neural Network-Oriented Indicator Method for Inverse Scattering Problems Using Partial Data," Mathematics, MDPI, vol. 12(4), pages 1-8, February.

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