LT-PINNs: Physics-informed neural networks based on Laplace transform for solving Caputo-type fractional partial differential equations

doi:10.1088/1674-1056/adf827

Abstract The solution of fractional partial differential equations (PDEs) is an important topic in scientific computing. However, the traditional physics-informed neural networks (PINNs) have problems of memory overflow and low computational efficiency when the derivative is discretized for a long time. Therefore in this paper we innovatively propose a framework of Laplace transform physics-informed neural networks (LT-PINNs), which is dedicated to solving the forward and inverse problems of Caputo-type fractional PDEs. The core of this method is to use the Laplace transform to construct the loss function, which skillfully avoids the dilemma that the fractional derivative operator in traditional PINNs is difficult to operate effectively. By studying the benchmark problem of parameter a in a series of different scenarios we verify that LT-PINNs can predict the solution of Caputo-type fractional PDEs more accurately than fractional PINNs. The excellent performance of LT-PINNs in identifying inverse problems involving fractional order, convection and diffusion coefficients is further explored. At the same time, the effects of network structure, the number of sampling points and noise on the LT-PINNs method are analyzed in detail. The results show that the method can predict the solution of the equation satisfactorily even under severe noise interference. The proposed LT-PINNs framework opens up a new path for efficiently solving fractional PDEs. It shows significant advantages in improving computational efficiency, reducing memory usage and dealing with complex noise environments. It is expected to promote the further development of fractional PDEs in many fields.

Keywords: fractional partial differential equation Caputo-type derivative physics-informed neural network Laplace transform

Received: 27 April 2025
Revised: 28 July 2025
Accepted manuscript online: 06 August 2025

PACS:	02.30.Jr	(Partial differential equations)
	07.05.Mh	(Neural networks, fuzzy logic, artificial intelligence)
	07.05.Tp	(Computer modeling and simulation)
	47.11.-j	(Computational methods in fluid dynamics)

Fund: This study was funded by the National Natural Science Foundation of China (Grant No. 12061055), the Key Projects of the Natural Science Foundation of Ningxia Hui Autonomous Region of China (Grant No. 2022AAC02005), and the Scientific and Technological Innovation Leading Talent Project of Ningxia Hui Autonomous Region of China (Grant No. 2021GKLRLX06).

Corresponding Authors: Fengjun Li, Jianqiang Liu
E-mail: fjli@nxu.edu.cn;liujq@amss.ac.cn

Cite this article:

Ruibo Zhang(张瑞波), Fengjun Li(李风军), and Jianqiang Liu(刘建强) LT-PINNs: Physics-informed neural networks based on Laplace transform for solving Caputo-type fractional partial differential equations 2026 Chin. Phys. B 35 030201

[1] Colbrook M and Ayton L 2022 J. Comput. Phys. 454 110995
[2] Lei D, Sun H, Zhang Y, Blaszczyk T and Yu Z 2023 J. Hydrol. 627 130280
[3] Chen L, Bajdich M, Martirez J, Krauter C, Gauthier J, Carter E, Luntz A, Chan K and Nørskov J 2018 Nat. Commun. 9 3202
[4] Fendzi-Donfack E, Temgoua G, Djoufack Z, Kenfack-Jiotsa A, Nguenang J and Nana L 2022 Chaos, Solitons and Fractals 160 112253
[5] Baleanu D, Shekari P, Torkzadeh L, Ranjbar H, Jajarmi A and Nouri K 2023 Chaos, Solitons and Fractals 166 112990
[6] Ngounou A, Feulefack S, Tabejieu L and Nbendjo B 2022 Chaos, Solitons and Fractals 157 111952
[7] Qing J, Zhou S, Wu J and Shao M 2023 Commun. Nonlinear Sci. Numer. Simul. 116 106810
[8] Hanif H 2021 Chaos, Solitons and Fractals 153 111463
[9] Feng L, Liu F, Turner I and Zhuang P 2017 Int. J. Heat Mass Tran. 115 1309
[10] Feng L, Liu F and Turner I 2019 Commun. Nonlinear Sci. Numer. Simul. 70 354
[11] Feng L, Liu F, Turner I and Zheng L 2018 Fract. Calc. Appl. Anal. 21 1073
[12] Chen X H, Yang W D, Zhang X R and Liu F W 2019 Appl. Math. Lett. 95 143
[13] Sheikh N, Ching D, Khan I and Sakidin H 2022 Sci. Rep. 12 14117
[14] Jani M, Babolian E and Bhatta D 2021 Math. Method. Appl. Sci. 44 2021
[15] Karipova G and Magdziarz M 2017 Chaos, Solitons and Fractals 102 245
[16] Kumar Y and Singh V 2021 Math. Comput. Simulat. 190 531
[17] Nuugulu S, Gideon F and Patidar 2021 Chaos, Solitons and Fractals 145 110753
[18] Wu H and Yang H 2026 Commun. Nonlinear Sci. Numer. Simul. 152 109133
[19] Podlubny I 1998 Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications (Amsterdam: Elsevier) Vol. 198
[20] Sun Z and Wu X 2006 Appl. Numer. Math. 56 193
[21] Lin Y and Xu C 2007 J. Comput. Phys. 225 1533
[22] Zhang Y, Sun Z and Wu H 2011 SIAM J. Numer. Anal. 49 2302
[23] Ren J and Sun Z 2013 J. Sci. Comput. 56 381
[24] Yang Q, Turner I, Liu F and Ilić M 2011 SIAM J. Sci. Comput. 33 1159
[25] Gao G, Sun Z and Zhang H 2014 J. Comput. Phys. 259 33
[26] Yang C 2023 Chaos, Solitons and Fractals 168 113204
[27] Xu H 2023 Chaos, Solitons and Fractals 166 112917
[28] Abdelrahim M, Saiga H, Maeda N, Hossain E, Ikeda H and Bhandari P 2022 Gut 71 7
[29] Cronin A, Smit J, Muñoz M, Poirier A, Moran P, Jerem P and Halfwerk W 2022 Biol. Rev. 97 1325
[30] Miller D, Stern A and Burstein D 2022 Nat. Commun. 13 5731
[31] Zhang R, Su J and Feng J 2024 Evol. Intell. 17 1567
[32] Adams S, Cody T and Beling P 2022 Artif. Intell. Rev. 55 4307
[33] Schneider J, Seidel S, Basalla M and vom Brocke J 2023 Bus. Inform. Syst. Eng. 65 65
[34] Brink A, Najera-Flores D and Martinez C 2021 Neural Comput. Appl. 33 5591
[35] Bar-Sinai Y, Hoyer S, Hickey J and Brenner M 2019 Proc. Natl. Acad. Sci. USA 116 15344
[36] Panghal S and Kumar M 2021 Eng. Comput. 37 2989
[37] Georgoulis E, Loulakis M and Tsiourvas A 2023 Commun. Nonlinear Sci. Numer. Simul. 117 106893
[38] Han J, Jentzen A and Weinan E 2018 Proc. Natl. Acad. Sci. USA 115 8505
[39] Raissi M, Perdikaris P and Karniadakis G 2017 arXiv:1711.10561 [cs.AI]
[40] Raissi M, Perdikaris P and Karniadakis G 2017 arXiv:1711.10566 [cs.AI]
[41] Raissi M, Perdikaris P and Karniadakis G 2019 J. Comput. Phys. 378 686
[42] Baydin A, Pearlmutter B, Radul A and Siskind J 2015 arXiv:1502.05767 [cs.SC]
[43] Jiang X, Wang D, Fan Q, Zhang M, Lu C and Lau A 2022 Laser Photonics Rev. 16 2100483
[44] Zhang R, Su J and Feng J 2023 Nonlinear Dynam. 111 13399
[45] Zhu Q, Liu Z and Yan J 2021 Comput. Mech. 67 619
[46] Goswami S, Anitescu C, Chakraborty S and Rabczuk T 2020 Theor. Appl. Fract. Mec. 106 102447
[47] Chen Y, Lu L, Karniadakis G and Dal Negro L 2020 Opt. Express 28 11618
[48] Bai Y, Chaolu T and Bilige S 2022 Nonlinear Dynam. 107 3655
[49] Mao Z, Jagtap A and Karniadakis G 2020 Comput. Methods Appl. Mech. Eng. 360 112789
[50] He Q, Barajas-Solano D, Tartakovsky G and Tartakovsky A 2020 Adv. Water. Resour. 141 103610
[51] Cai S, Mao Z, Wang Z, Yin M and Karniadakis G 2021 Acta. Mech. Sin. 37 1727
[52] Pang G, Lu L and Karniadakis G 2019 SIAM J. Sci. Comput. 41 A2603
[53] Guo L, Wu H, Yu X and Zhou T 2022 Comput. Methods Appl. Mech. Eng. 400 115523
[54] Zhang T, Zhang D, Shi S and Guo Z 2025 Nonlinear Dynam. 113 12565
[55] Ren J, Sun Z and Dai W 2016 Appl. Math. Model. 40 2625
[56] Hornik K, Stinchcombe M and White H 1990 Neural networks 3 551
[57] Kilbas A, Srivastava H and Trujillo J Theory and applications of fractional differential equations (Amsterdam: Elsevier) Vol. 204
[58] Kingma D and Ba J 2014 arXiv:1412.6980 [cs.ML]
[59] Liu D and Nocedal J 1989 Math. Program. 45 503
[60] McKay M, Beckman R and Conover W 2000 Technometrics 42 55

[1]	TCAS-PINN: Physics-informed neural networks with a novel temporal causality-based adaptive sampling method Jia Guo(郭嘉), Haifeng Wang(王海峰), Shilin Gu(古仕林), and Chenping Hou(侯臣平). Chin. Phys. B, 2024, 33(5): 050701.
[2]	MetaPINNs: Predicting soliton and rogue wave of nonlinear PDEs via the improved physics-informed neural networks based on meta-learned optimization Yanan Guo(郭亚楠), Xiaoqun Cao(曹小群), Junqiang Song(宋君强), and Hongze Leng(冷洪泽). Chin. Phys. B, 2024, 33(2): 020203.
[3]	Meshfree-based physics-informed neural networks for the unsteady Oseen equations Keyi Peng(彭珂依), Jing Yue(岳靖), Wen Zhang(张文), and Jian Li(李剑). Chin. Phys. B, 2023, 32(4): 040207.
[4]	Adaptive multi-step piecewise interpolation reproducing kernel method for solving the nonlinear time-fractional partial differential equation arising from financial economics Ming-Jing Du(杜明婧), Bao-Jun Sun(孙宝军), and Ge Kai(凯歌). Chin. Phys. B, 2023, 32(3): 030202.
[5]	Compact finite difference schemes for the backward fractional Feynman-Kac equation with fractional substantial derivative Jiahui Hu(胡嘉卉), Jungang Wang(王俊刚), Yufeng Nie(聂玉峰), Yanwei Luo(罗艳伟). Chin. Phys. B, 2019, 28(10): 100201.
[6]	The bound state solution for the Morse potential with a localized mass profile S Miraboutalebi. Chin. Phys. B, 2016, 25(10): 100301.
[7]	Relativistic effect of pseudospin symmetry and tensor coupling on the Mie-type potential via Laplace transformation method M. Eshghi, S. M. Ikhdair. Chin. Phys. B, 2014, 23(12): 120304.
[8]	New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology M. M. Khader, Sunil Kumar, S. Abbasbandy. Chin. Phys. B, 2013, 22(11): 110201.
[9]	Variational iteration method for solving the time-fractional diffusion equations in porous medium Wu Guo-Cheng (吴国成). Chin. Phys. B, 2012, 21(12): 120504.
[10]	Element-free Galerkin (EFG) method for analysis of the time-fractional partial differential equations Ge Hong-Xia(葛红霞), Liu Yong-Qing(刘永庆), and Cheng Rong-Jun(程荣军) . Chin. Phys. B, 2012, 21(1): 010206.
[11]	Dynamic analysis of a new chaotic system with fractional order and its generalized projective synchronization Niu Yu-Jun(牛玉军), Wang Xing-Yuan(王兴元), Nian Fu-Zhong(年福忠), and Wang Ming-Jun(王明军). Chin. Phys. B, 2010, 19(12): 120507.
[12]	Exact solutions of N-dimensional harmonic oscillator via Laplace transformation Chen Gang (陈刚). Chin. Phys. B, 2005, 14(6): 1075-1076.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

LT-PINNs: Physics-informed neural networks based on Laplace transform for solving Caputo-type fractional partial differential equations

Cite this article:

Online attention