Gradient-enhanced fractional physics-informed neural networks for solving forward and inverse problems of the multiterm time-fractional Burger-type equation

Shanhao Yuan; Yanqin Liu; Yibin Xu; Qiuping Li; Chao Guo; Yanfeng Shen; Shanhao Yuan; Yanqin Liu; Yibin Xu; Qiuping Li; Chao Guo; Yanfeng Shen

doi:10.3934/math.20241332

AIMS Mathematics

2024, Volume 9, Issue 10: 27418-27437. doi: 10.3934/math.20241332

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Gradient-enhanced fractional physics-informed neural networks for solving forward and inverse problems of the multiterm time-fractional Burger-type equation

1.
School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353, China
2.
School of Mathematics and Big Data, Dezhou University, Dezhou 253023, China
3.
Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China

Received: 15 July 2024 Revised: 05 September 2024 Accepted: 18 September 2024 Published: 23 September 2024
MSC : 26A33, 35R11, 65M22, 68T07

In this paper, we introduced the gradient-enhanced fractional physics-informed neural networks (gfPINNs) for solving the forward and inverse problems of the multiterm time-fractional Burger-type equation. The gfPINNs leverage gradient information derived from the residual of the fractional partial differential equation and embed the gradient into the loss function. Since the standard chain rule in integer calculus is invalid in fractional calculus, the automatic differentiation of neural networks does not apply to fractional operators. The automatic differentiation for the integer order operators and the finite difference discretization for the fractional operators were used to construct the residual in the loss function. The numerical results demonstrate the effectiveness of gfPINNs in solving the multiterm time-fractional Burger-type equation. By comparing the experimental results of fractional physics-informed neural networks (fPINNs) and gfPINNs, it can be seen that the training performance of gfPINNs is better than fPINNs.
- multiterm time-fractional Burger-type equation,
- fPINNs,
- gfPINNs,
- forward problem,
- inverse problem
Citation: Shanhao Yuan, Yanqin Liu, Yibin Xu, Qiuping Li, Chao Guo, Yanfeng Shen. Gradient-enhanced fractional physics-informed neural networks for solving forward and inverse problems of the multiterm time-fractional Burger-type equation[J]. AIMS Mathematics, 2024, 9(10): 27418-27437. doi: 10.3934/math.20241332

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Abstract

In this paper, we introduced the gradient-enhanced fractional physics-informed neural networks (gfPINNs) for solving the forward and inverse problems of the multiterm time-fractional Burger-type equation. The gfPINNs leverage gradient information derived from the residual of the fractional partial differential equation and embed the gradient into the loss function. Since the standard chain rule in integer calculus is invalid in fractional calculus, the automatic differentiation of neural networks does not apply to fractional operators. The automatic differentiation for the integer order operators and the finite difference discretization for the fractional operators were used to construct the residual in the loss function. The numerical results demonstrate the effectiveness of gfPINNs in solving the multiterm time-fractional Burger-type equation. By comparing the experimental results of fractional physics-informed neural networks (fPINNs) and gfPINNs, it can be seen that the training performance of gfPINNs is better than fPINNs.

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