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Explore deep network for a class of fractional partial differential equations

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  • Fang, Xing
  • Qiao, Leijie
  • Zhang, Fengyang
  • Sun, Fuming

Abstract

In this paper, we present a novel approach for solving a class of fractional partial differential equations (FPDEs) and their inverse problems using deep neural networks (DNNs). Our proposed framework utilizes the discrete Caputo fractional derivative method to approximate fractional partial derivatives, while leveraging automatic differentiation of neural networks to obtain integer derivatives. This approach offers several advantages, including avoiding the direct solution of the original FPDEs and overcoming the limitations faced by traditional numerical methods in handling FPDEs. To validate our approach, we provide numerical examples with known analytical solutions, accompanied by graphical and numerical results. Our findings demonstrate that the proposed method is easily implementable, exhibits fast convergence, robustness, and effectiveness in solving multidimensional FPDEs and their inverse problems.

Suggested Citation

  • Fang, Xing & Qiao, Leijie & Zhang, Fengyang & Sun, Fuming, 2023. "Explore deep network for a class of fractional partial differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 172(C).
    • Handle: RePEc:eee:chsofr:v:172:y:2023:i:c:s0960077923004290
    DOI: 10.1016/j.chaos.2023.113528
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    References listed on IDEAS

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    1. Qiao, Leijie & Xu, Da & Wang, Zhibo, 2019. "An ADI difference scheme based on fractional trapezoidal rule for fractional integro-differential equation with a weakly singular kernel," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 103-114.
    2. Qu, Haidong & She, Zihang & Liu, Xuan, 2021. "Neural network method for solving fractional diffusion equations," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    3. Xu, Da, 2017. "Numerical asymptotic stability for the integro-differential equations with the multi-term kernels," Applied Mathematics and Computation, Elsevier, vol. 309(C), pages 107-132.
    4. John Rust, 1997. "Using Randomization to Break the Curse of Dimensionality," Econometrica, Econometric Society, vol. 65(3), pages 487-516, May.
    5. Owyed, Saud & Abdou, M.A. & Abdel-Aty, Abdel-Haleem & Alharbi, W. & Nekhili, Ramzi, 2020. "Numerical and approximate solutions for coupled time fractional nonlinear evolutions equations via reduced differential transform method," Chaos, Solitons & Fractals, Elsevier, vol. 131(C).
    6. Laiq Zada & Rashid Nawaz & Mohammad A. Alqudah & Kottakkaran Sooppy Nisar, 2022. "A New Technique For Approximate Solution Of Fractional-Order Partial Differential Equations," FRACTALS (fractals), World Scientific Publishing Co. Pte. Ltd., vol. 30(01), pages 1-8, February.
    7. Wubshet Ibrahim & Lelisa Kebena Bijiga, 2021. "Neural Network Method for Solving Time-Fractional Telegraph Equation," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-10, May.
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    Cited by:

    1. Yang, Junxiang & Lee, Dongsun & Kwak, Soobin & Ham, Seokjun & Kim, Junseok, 2024. "The Allen–Cahn model with a time-dependent parameter for motion by mean curvature up to the singularity," Chaos, Solitons & Fractals, Elsevier, vol. 182(C).
    2. Ma, Zhiying & Hou, Jie & Zhu, Wenhao & Peng, Yaxin & Li, Ying, 2023. "PMNN: Physical model-driven neural network for solving time-fractional differential equations," Chaos, Solitons & Fractals, Elsevier, vol. 177(C).

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