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Solving Benjamin–Ono equation via gradient balanced PINNs approach

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Abstract

Soliton solution of Benjamin–Ono (BO) equation has considerable applications in physical field. In this paper, we utilize the Physics-informed neural networks (PINNs) to solve the soliton solution of the BO equation firstly. However, we discern that the imbalance phenomenon of back-propagated gradients occurs between residual loss and initial values or boundary conditions, which leads to the loss function failing to converge well during model training. In order to overcome the faultiness, we introduce self-adaptive parameters into the loss function in PINNs to balance the effect between different terms in original loss function, namely Gradient Balanced PINNs approach. By comparing the performance of solving BO equation, numerical results showed that the prediction accuracy of improved method has improved significantly and absolute error is reduced by order of magnitude. Finally, relevant results have been exhibited in tables and diagrams, dynamical behaviors and error analysis by employing two methods are revealed in detail.

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Data availability statement

This manuscript has associated data in a data repository. [Authors’comment: The datasets generated or analyzed during the current study are available from the corresponding author on reasonable request.]

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Acknowledgements

Project supported by National Natural Science Foundation of China (Grant Nos. 52171251, U2106225), LiaoNing Revitalization Talents Program (XLYC1907014) and the Fundamental Research Funds for the Central Universities (DUT21ZD205).

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  1. School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Xiangyu Yang & Zhen Wang

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  1. Xiangyu Yang
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Correspondence to Zhen Wang.

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Yang, X., Wang, Z. Solving Benjamin–Ono equation via gradient balanced PINNs approach. Eur. Phys. J. Plus 137, 864 (2022). https://doi.org/10.1140/epjp/s13360-022-03078-8

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