中国物理B ›› 2026, Vol. 35 ›› Issue (3): 30201-030201.doi: 10.1088/1674-1056/adf827

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

Ruibo Zhang(张瑞波)1,3, Fengjun Li(李风军)2,3,†, and Jianqiang Liu(刘建强)1,3,‡   

  1. 1 School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China;
    2 School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China;
    3 Ningxia Basic Science Research Center of Mathematics, Ningxia University, Yinchuan 750021, China
  • 收稿日期:2025-04-27 修回日期:2025-07-28 接受日期:2025-08-06 发布日期:2026-02-11
  • 基金资助:
    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).

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

Ruibo Zhang(张瑞波)1,3, Fengjun Li(李风军)2,3,†, and Jianqiang Liu(刘建强)1,3,‡   

  1. 1 School of Mathematics and Statistics, Ningxia University, Yinchuan 750021, China;
    2 School of Mathematics and Information Science, North Minzu University, Yinchuan 750021, China;
    3 Ningxia Basic Science Research Center of Mathematics, Ningxia University, Yinchuan 750021, China
  • Received:2025-04-27 Revised:2025-07-28 Accepted:2025-08-06 Published:2026-02-11
  • Contact: Fengjun Li, Jianqiang Liu E-mail:fjli@nxu.edu.cn;liujq@amss.ac.cn
  • Supported by:
    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).

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摘要： 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.

关键词: fractional partial differential equation, Caputo-type derivative, physics-informed neural network, Laplace transform

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.

Key words: fractional partial differential equation, Caputo-type derivative, physics-informed neural network, Laplace transform

中图分类号:  (Partial differential equations)

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