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Data-driven forward–inverse problems for the variable coefficients Hirota equation using deep learning method

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Abstract

This paper investigates data-driven forward–inverse problems associated with the variable coefficients Hirota (VC-Hirota) equation using the physics-informed neural network (PINN) algorithm. First, we propose an improved PINN algorithm with a locally adaptive activation function to recover data-driven solitons and high-order solitons solutions for the VC-Hirota equation. Second, we demonstrate the effectiveness of the improved PINN algorithm in accurately predicting parameters under different noise intensities using a parameter regularization strategy and appropriate weight coefficients. Third, we introduce a PINNs approach that employs two neural networks to tackle the function discovery problem. The neural network with time and space coordinates in the input layer is used to train the prediction solution, and the neural network with only time coordinates in the input layer is trained to model the unknown function in the variable coefficient function. This work presents a successful attempt to use the PINN method to solve the function discovery problem of VC-Hirota equations.

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Acknowledgements

The authors would like to thank Zhengwu Miao sincerely for providing with valuable comments.

Funding

This work is supported by National Natural Science Foundation of China (No.12175069 and No.12235007), Science and Technology Commission of Shanghai Municipality (No.21JC1402500 and No.22DZ2229014), and Natural Science Foundation of Shanghai (No.23ZR1418100).

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Authors and Affiliations

  1. School of Mathematical Sciences, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241, People’s Republic of China

    Huijuan Zhou, Juncai Pu & Yong Chen

  2. College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao, 266590, People’s Republic of China

    Yong Chen

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  1. Huijuan Zhou
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Correspondence to Yong Chen.

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The authors would like to thank Zhengwu Miao sincerely for providing with valuable comments. This work is supported by National Natural Science Foundation of China (No. 12175069 and No. 12235007) and Science and Technology Commission of Shanghai Municipality (No. 21JC1402500 and No. 22DZ2229014).

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Zhou, H., Pu, J. & Chen, Y. Data-driven forward–inverse problems for the variable coefficients Hirota equation using deep learning method. Nonlinear Dyn 111, 14667–14693 (2023). https://doi.org/10.1007/s11071-023-08641-1

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