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Solving Huxley equation using an improved PINN method

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Abstract

Although many effective methods for solving partial differential equations (PDEs) have been proposed, there is no universal method that can solve all PDEs. Therefore, solving partial differential equations has always been a difficult problem in mathematics, such as deep neural network (DNN). In recent years, a method of embedding some basic physical laws into traditional neural networks has been proposed to reveal the dynamic behavior of equations directly from space-time data [i.e., physics-informed neural network (PINN)]. Based on the above, an improved deep learning method to recover the new soliton solution of Huxley equation has been proposed in this paper. As far as we know, this is the first time that we have used an improved method to study the numerical solution of the Huxley equation. In order to illustrate the advantages of the improved method, we use the same network depth, the same hidden layer and neurons contained in the hidden layer, and the same training sample points. We analyze the dynamic behavior and error of Huxley’s exact solution and the new soliton solution and give vivid graphs and detailed analysis. Numerical results show that the improved algorithm can use fewer sample points to reconstruct the exact solution of the Huxley equation with faster convergence speed and better simulation effect.

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Data availability statement

The data sets generated or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work is supported by National Natural Science Foundation of China (11571008), (11661060), and (12061054) and Natural Science Foundation of Inner Mongolia Autonomous Region of China (2018LH01013).

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Authors and Affiliations

  1. College of Information Engineering, Shanghai Maritime University, Shanghai, 201306, China

    Yuexing Bai

  2. College of Arts and Sciences, Shanghai Maritime University, Shanghai, 201306, China

    Temuer Chaolu

  3. Department of Mathematics, Inner Mongolia University of Technology, Hohhot, 010051, China

    Sudao Bilige

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  1. Yuexing Bai
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  2. Temuer Chaolu
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  3. Sudao Bilige
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Correspondence to Yuexing Bai.

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Bai, Y., Chaolu, T. & Bilige, S. Solving Huxley equation using an improved PINN method. Nonlinear Dyn 105, 3439–3450 (2021). https://doi.org/10.1007/s11071-021-06819-z

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