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<p>This paper investigates a nonlinear inverse problem of identifying simultaneously the order of time-fractional derivative and the space-dependent potential coefficient in a one-dimensional time-fractional diffusion-wave equation from the lateral Cauchy data. The existence and uniqueness of weak solution for the direct problem with non-homogeneous boundary conditions are discussed. Based on the weak solution of the direct problem, the uniqueness for the simultaneous determination of the fractional order and the space-dependent potential coefficient is proved by analytic continuation, Laplace transformation and Gel’fand–Levitan theory under some suitable conditions. Moreover, we employ an Adaptive fractional Physics-Informed Neural Networks (Adaptive-fPINNs) for numerical reconstructions of the fractional order and the space-dependent potential coefficient simultaneously. The numerical experimental results for three examples fully indicate the rationality of the theory and the effectiveness of the numerical method.</p>

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Simultaneous identification of the order and potential coefficient in a time fractional diffusion-wave equation by a deep neural networks method

  • 期刊论文

  • 发表日期：

    2025年6月28日

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  • Yuhua Luo,
  • Mingjia Pei,
  • Ting Wei

摘要

This paper investigates a nonlinear inverse problem of identifying simultaneously the order of time-fractional derivative and the space-dependent potential coefficient in a one-dimensional time-fractional diffusion-wave equation from the lateral Cauchy data. The existence and uniqueness of weak solution for the direct problem with non-homogeneous boundary conditions are discussed. Based on the weak solution of the direct problem, the uniqueness for the simultaneous determination of the fractional order and the space-dependent potential coefficient is proved by analytic continuation, Laplace transformation and Gel’fand–Levitan theory under some suitable conditions. Moreover, we employ an Adaptive fractional Physics-Informed Neural Networks (Adaptive-fPINNs) for numerical reconstructions of the fractional order and the space-dependent potential coefficient simultaneously. The numerical experimental results for three examples fully indicate the rationality of the theory and the effectiveness of the numerical method.