Efficient Legendre Polynomial Neural Networks Method for Time-Fractional Partial Differential Equations with Singularity

doi:10.1007/s42967-025-00509-y

Communications on Applied Mathematics and Computation ›› 2026, Vol. 8 ›› Issue (4): 1527-1549.doi: 10.1007/s42967-025-00509-y

Efficient Legendre Polynomial Neural Networks Method for Time-Fractional Partial Differential Equations with Singularity

Hong-Liang Huang¹, Da-Kang Cen², Seak-Weng Vong¹, Siu-Long Lei¹

1. Department of Mathematics, University of Macau, Macau, China;
2. Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong, China

收稿日期:2025-04-10 修回日期:2025-05-23 接受日期:2025-06-02 出版日期:2026-08-20 发布日期:2026-09-17
通讯作者: Siu-Long Lei, Email: sllei@um.edu.mo E-mail:sllei@um.edu.mo
基金资助:
This work is supported in part by research grants from the University of Macau (file nos. MYRG-GRG2023-00181-FST-UMDF, MYRG-GRG2024-00237-FST-UMDF, and MYRG-GRG2024-00100-FST-UMDF) and the Science and Technology Development Fund, Macau SAR (file no. 0151/2022/A). We would like to express our gratitude to Dr. Huang Yunchi from the Department of Mathematics at Shantou University for his guidance.

Efficient Legendre Polynomial Neural Networks Method for Time-Fractional Partial Differential Equations with Singularity

Hong-Liang Huang¹, Da-Kang Cen², Seak-Weng Vong¹, Siu-Long Lei¹

1. Department of Mathematics, University of Macau, Macau, China;
2. Department of Mathematics, Southern University of Science and Technology, Shenzhen, Guangdong, China

Received:2025-04-10 Revised:2025-05-23 Accepted:2025-06-02 Online:2026-08-20 Published:2026-09-17
Supported by:
This work is supported in part by research grants from the University of Macau (file nos. MYRG-GRG2023-00181-FST-UMDF, MYRG-GRG2024-00237-FST-UMDF, and MYRG-GRG2024-00100-FST-UMDF) and the Science and Technology Development Fund, Macau SAR (file no. 0151/2022/A). We would like to express our gratitude to Dr. Huang Yunchi from the Department of Mathematics at Shantou University for his guidance.

摘要/Abstract

摘要： Compared to multi-layer neural networks based on nonlinear activation functions, single-layer neural networks that use smooth linear functions as activation functions can quickly numerically solve time-fractional partial differential equations (TFPDEs) with time derivative of order α ∈ (0, 1). However, due to the weak singularity of the solution which is not smooth near the initial time, resulting in significant numerical errors when using neural networks based on smooth functions as activation functions. In this paper, we develop a decomposition method to reduce the singularity of the initial value, and theoretically demonstrate that this method can reduce the singularity. Then, a new single-layer neural network and a fast training method are proposed based on the decomposition method and the smooth linear activation function. Numerical experiments have shown that this method reduces errors and improves computational speed.

关键词: Singularity, Caputo fractional derivative, Caputo-Hadamard fractional derivative, Fitted method, Neural networks

Abstract: Compared to multi-layer neural networks based on nonlinear activation functions, single-layer neural networks that use smooth linear functions as activation functions can quickly numerically solve time-fractional partial differential equations (TFPDEs) with time derivative of order α ∈ (0, 1). However, due to the weak singularity of the solution which is not smooth near the initial time, resulting in significant numerical errors when using neural networks based on smooth functions as activation functions. In this paper, we develop a decomposition method to reduce the singularity of the initial value, and theoretically demonstrate that this method can reduce the singularity. Then, a new single-layer neural network and a fast training method are proposed based on the decomposition method and the smooth linear activation function. Numerical experiments have shown that this method reduces errors and improves computational speed.

Key words: Singularity, Caputo fractional derivative, Caputo-Hadamard fractional derivative, Fitted method, Neural networks

Hong-Liang Huang, Da-Kang Cen, Seak-Weng Vong, Siu-Long Lei. Efficient Legendre Polynomial Neural Networks Method for Time-Fractional Partial Differential Equations with Singularity[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1527-1549.

[1] Brunner, H., Han, H., Yin, D.: The maximum principle for time-fractional diffusion equations and its application. Numer. Funct. Anal. Optim. 36(10), 1307-1321 (2015)
[2] Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional-in-space reaction-diffusion equations. BIT Numer. Math. 54(4), 937-954 (2014)
[3] Burrage, K., Hale, N., Kay, D.: An efficient implicit FEM scheme for fractional-in-space reaction-diffusion equations. SIAM J. Sci. Comput. 34(4), A2145-A2172 (2012)
[4] Cen, D., Vong, S.: The tracking of derivative discontinuities for delay fractional equations based on fitted L1 method. Comput. Meth. Appl. Mat. 23(3), 591-601 (2023)
[5] Courant, R., Hilbert, D.: Methods of Mathematical Physics. Interscience Publishers, New York (1953)
[6] Doha, E., Bhrawy, A., Ezz-Eldien, S.: A new Jacobi operational matrix: an application for solving fractional differential equations. Appl. Math. Model. 36(10), 4931-4943 (2012)
[7] Doha, E., Bhrawy, A., Hafez, R.: On shifted Jacobi spectral method for high-order multi-point boundary value problems. Commun. Nonlinear Sci. Numer. Simul. 17(10), 3802-3810 (2012)
[8] Du, N., Wang, H.: A fast finite element method for space-fractional dispersion equations on bounded domains in \mathbb{R} ^2. SIAM J. Sci. Comput. 37(3), A1614-A1635 (2015)
[9] Ford, N.J., Xiao, J., Yan, Y.: A finite element method for time fractional partial differential equations. Fract. Calc. Appl. Anal. 14, 454-474 (2011)
[10] Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions. Related Topics and Applications. Springer, Germany (2014)
[11] Gracia, J.L., O’Riordan, E., Stynes, M.: A fitted scheme for a Caputo initial-boundary value problem. J. Sci. Comput. 76, 583-609 (2018)
[12] Guo, L., Wu, H., Yu, X., Zhou, T.: Monte Carlo PINNs: deep learning approach for forward and inverse problems involving high dimensional fractional partial differential equations. https://doi.org/10.48550/arXiv.2203.08501 (2022)
[13] Hao, Z., Moongyu, P., Cai, Z.: Neural network method for integral fractional Laplace equations. East Asian J. Appl. Math. 13, 95-118 (2023)
[14] Hornik, K.: Approximation capabilities of multilayer feedforward networks. Neural Netw. 4(2), 251-257 (1991)
[15] Ilic, M., Liu, F., Turner, I., Anh, V.: Numerical approximation of a fractional-in-space diffusion equation. I. Fract. Calc. Appl. Anal. 8(3), 323-341 (2005)
[16] Jarad, F., Abdeljawad, T., Baleanu, D.: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 142, 1-8 (2012)
[17] Jiang, Y., Ma, J.: High-order finite element methods for time-fractional partial differential equations. J. Comput. Appl. Math. 235(11), 3285-3290 (2011)
[18] Ke, R., Ng, M., Sun, H.: A fast direct method for block triangular Toeplitz-like with tri-diagonal block systems from time-fractional partial differential equations. J. Comput. Phys. 303, 203-211 (2015)
[19] Li, C., Li, Z., Wang, Z.: Mathematical analysis and the local discontinuous Galerkin method for Caputo-Hadamard fractional partial differential equation. J. Sci. Comput. 85, 1-27 (2020)
[20] Lin, X., Ng, M., Sun, H.: A splitting preconditioner for Toeplitz-like linear systems arising from fractional diffusion equations. SIAM J. Matrix Anal. Appl. 38(4), 1580-1614 (2017)
[21] Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space-time fractional advection-diffusion equation. Appl. Math. Comput. 191(1), 12-20 (2007)
[22] Liu, M., Cai Z., Adaptive two-layer ReLU neural network: II. Ritz approximation to elliptic PDEs. Comput. Math. Appl. 113, 103-116 (2022)
[23] Lu, L., Meng, X., Mao, Z., Karniadakis, G.E.: DeepXDE: a deep learning library for solving differential equations. SIAM Rev. 63(1), 208-228 (2019)
[24] Ma, L., Li, R., Zeng, F., Karniadakis, G.E.: Bi-orthogonal fPINN: a physics-informed neural network method for solving time-dependent stochastic fractional PDEs. Commun. Comput. Phys. 34(4), 1133-1176 (2023)
[25] Ma, Z., Hou J., Zhu W., Peng Y., Li Y.: PMNN: physical model-driven neural network for solving time-fractional differential equations. Chaos Solitons Fractals 177, 114238 (2023)
[26] Meerschaert, M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion flow equations. J. Comput. Appl. Math. 172(1), 65-77 (2004)
[27] Meerschaert, M., Tadjeran, C.: Finite difference approximations for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56(1), 80-90 (2006)
[28] Ou, C., Cen, D., Vong, S., Wang, Z.: Mathematical analysis and numerical methods for Caputo-Hadamard fractional diffusion-wave equations. Appl. Numer. Math. 177, 34-57 (2022)
[29] Ou, C., Cen, D., Wang, Z., Vong, S.: Fitted schemes for Caputo-Hadamard fractional differential equations. Numer. Algorithms 97(1), 1-30 (2023)
[30] Pang, G., Lu, L., Karniadakis, G.E.: fPINNs: fractional physics-informed neural networks. SIAM J. Sci. Comput. 41(4), A2603-A2626 (2019)
[31] Podlubny, I.: Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Academic Press, San Diego (1999)
[32] Qu, H., Liu, X., Lu, X., Ur Rahman, M., She, Z.: Neural network method for solving nonlinear fractional advection-diffusion equation with spatiotemporal variable-order. Chaos Solitons Fractals 156, 111856 (2022)
[33] Qu, H., Liu, X., She, Z.: Neural network method for fractional-order partial differential equations. Neurocomputing 414, 225-237 (2020)
[34] Qu, H., She, Z., Liu, X.: Neural network method for solving fractional diffusion equations. Appl. Math. Comput. 391(1), 125635 (2021)
[35] Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686-707 (2018)
[36] Roop, J., John, P.: Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in \mathbb{R} ^2. J. Comput. Appl. Math. 193(1), 243-268 (2006)
[37] Sakamoto, K., Yamamoto, M.: Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems. J. Math. Anal. Appl. 382(1), 426-447 (2011)
[38] Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms. Analysis and Applications. Springer, Germany (2011)
[39] Sivalingam, S.M., Govindaraj, V.: Physics informed neural network based scheme and its error analysis for \psi-Caputo type fractional differential equations. Phys. Scripta 99(9), 096002 (2024)
[40] Stynes, M., O’Riordan, E., Gracia, J.L.: Error analysis of a finite difference method on graded meshes for a time-fractional diffusion equation. SIAM J. Numer. Anal. 55(2), 1057-1079 (2017)
[41] Sugandha, A., Trilok, M., Shivi, A., Kamlesh, T., Phalguni, G.: Applications of fractional calculus in computer vision: a survey. Neurocomputing 489, 407-428 (2022)
[42] Tian, W., Zhou, H., Deng, W.: A class of second order difference approximations for solving space fractional diffusion equations. Math. Comput. 84, 1703-1727 (2015)
[43] Wang, Q., Ma, J., Yu, S., Tan, L.: Noise detection and image denoising based on fractional calculus. Chaos Solitons Fractals 131, 109463 (2020)
[44] Yadav, P., Jahan, S., Nisar, K.S.: Solving fractional Bagley-Torvik equation by fractional order Fibonacci wavelet arising in fluid mechanics. Ain Shams Eng. J. 15(1), 102299 (2024)
[45] Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34(1), 200-218 (2010)
[46] Ye, Y., Li, Y., Fan, H., Liu, X., Zhang, H.: SLeNN-ELM: a shifted Legendre neural network method for fractional delay differential equations based on extreme learning machine. Netw. Heterog. Media 18(1), 494-512 (2023)
[47] Zayernouri, M., Karniadakis, G.E.: Fractional spectral collocation method. SIAM J. Sci. Comput. 36(1), A40-A62 (2014)
[48] Zeng, F., Liu, F., Li, C.: A Crank-Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. SIAM J. Numer. Anal. 52(6), 2599-2622 (2014)
[49] Zhao, T., Li, C., Li, D.: Efficient spectral collocation method for fractional differential equation with Caputo-Hadamard derivative. Fract. Calc. Appl. Anal. 26, 2903-2927 (2023)

Efficient Legendre Polynomial Neural Networks Method for Time-Fractional Partial Differential Equations with Singularity

Efficient Legendre Polynomial Neural Networks Method for Time-Fractional Partial Differential Equations with Singularity

PDF

可视化

摘要/Abstract

引用本文

使用本文

相关文章 15

编辑推荐

Metrics

本文评价

[1]	Jostein Barry-Straume, Arash Sarshar, Andrey A. Popov, Adrian Sandu. Physics-Informed Neural Networks for PDE-Constrained Optimization and Control[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1283-1306.
[2]	Ya-ru Liu, Guang-hua Gao, Jun-zhen Lu. Exploration of Two Modified L1 Formulae to Approximate the Caputo Fractional Derivative[J]. Communications on Applied Mathematics and Computation, 2026, 8(4): 1550-1586.
[3]	Haili Qiao, Aijie Cheng. A Fast Averaged L1 Finite Difference Method for Time Fractional Mobile/Immobile Diffusion Equation with Weakly Singular Solution[J]. Communications on Applied Mathematics and Computation, 2026, 8(3): 1207-1225.
[4]	Derk Frerichs-Mihov, Linus Henning, Volker John. On Loss Functionals for Physics-Informed Neural Networks for Steady-State Convection-Dominated Convection-Diffusion Problems[J]. Communications on Applied Mathematics and Computation, 2026, 8(1): 287-308.
[5]	Nikki Kedia, Anatoly A. Alikhanov, Vineet Kumar Singh. Higher Order Computational Approach for Generalized Time-Fractional Diffusion Equation[J]. Communications on Applied Mathematics and Computation, 2025, 7(6): 2462-2484.
[6]	A. M. Vargas. A Finite Difference Scheme for the Fractional Laplacian on Non-uniform Grids[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1364-1377.
[7]	T. Nasiri, A. Zakeri, A. Aminataei. Quasi Solution of an Inverse Fractional Stochastic Nonlinear Partial Differential Equation of Parabolic Type[J]. Communications on Applied Mathematics and Computation, 2025, 7(4): 1350-1363.
[8]	Juan Ruiz-álvarez, Baiying Dong, Zhilin Li. The Immersed Interface Method for Navier-Stokes Equations with Interfaces in Cylindrical Coordinates[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 1074-1097.
[9]	Baoli Yin, Yang Liu, Hong Li. Sharp Error Analysis for Averaging Crank-Nicolson Schemes with Corrections for Subdiffusion with Nonsmooth Solutions[J]. Communications on Applied Mathematics and Computation, 2025, 7(3): 865-884.
[10]	Jarrod Mau, Jia Zhao. Discovery of Governing Equations with Recursive Deep Neural Networks[J]. Communications on Applied Mathematics and Computation, 2025, 7(1): 239-263.
[11]	Jim Magiera, Christian Rohde. A Multiscale Method for Two-Component, Two-Phase Flow with a Neural Network Surrogate[J]. Communications on Applied Mathematics and Computation, 2024, 6(4): 2265-2294.
[12]	Zhiwei Gao, Tao Tang, Liang Yan, Tao Zhou. Failure-Informed Adaptive Sampling for PINNs, Part II: Combining with Re-sampling and Subset Simulation[J]. Communications on Applied Mathematics and Computation, 2024, 6(3): 1720-1741.
[13]	Zhanhong Ye, Xiang Huang, Hongsheng Liu, Bin Dong. Meta-Auto-Decoder: a Meta-Learning-Based Reduced Order Model for Solving Parametric Partial Differential Equations[J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 1096-1130.
[14]	Jane Wu, Michael Bao, Xinwei Yao, Ronald Fedkiw. Deep Energies for Estimating Three-Dimensional Facial Pose and Expression[J]. Communications on Applied Mathematics and Computation, 2024, 6(2): 837-861.
[15]	Zhengnan Dong, Enyu Fan, Ao Shen, Yuhao Su. Three Kinds of Discrete Formulae for the Caputo Fractional Derivative[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1446-1468.