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Data-Driven Deep Learning for The Multi-Hump Solitons and Parameters Discovery in NLS Equations with Generalized \({\mathcal{PT}\mathcal{}}\)-Scarf-II Potentials

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Abstract

In this paper, we investigate the data-driven forward and inverse problems of both the focusing and defocusing nonlinear Schrödinger equations (NLSEs) with generalized parity-time (\({\mathcal{PT}\mathcal{}}\))-Scarf-II potential via the physics-informed neural networks (PINNs) deep learning. The NLSE with four different initial conditions and periodic boundary condition are analyzed via the PINNs approach. And the predicted (data-driven) multi-hump solitons have been compared to the solutions, which can be obtained from the analytical or the high-accuracy numerical methods. Moreover, we explore the influences of several key factors (e.g., the depth of the neural networks, activation functions) on the performance of the PINNs algorithm. Finally, the data-driven inverse problems of the NLSE are also investigated such that the coefficients of the generalized \({\mathcal{PT}\mathcal{}}\)-Scarf-II potentials, the nonlinear and dispersion terms can be found. The results obtained in this paper can be used to further explore the NLSE with \({\mathcal{PT}\mathcal{}}\)-symmetric potentials and the applications of deep learning method in the nonlinear partial differential equations.

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Acknowledgements

The work of Z.Y. was supported by the NSFC under Grant Nos. 11925108 and 11731014. The work of S-F.T. was supported by the NSFC under Grant No. 11975306, the NSF of Jiangsu Province of China (Grant No. BK20181351) and the the Six Talent Peaks Project in Jiangsu Province (Grant No. JY-059).

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Authors and Affiliations

  1. KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

    Ming Zhong, Zijian Zhou & Zhenya Yan

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China

    Ming Zhong, Zijian Zhou & Zhenya Yan

  3. School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China

    Jian-Guo Zhang & Shou-Fu Tian

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  1. Ming Zhong
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Correspondence to Zhenya Yan.

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Zhong, M., Zhang, JG., Zhou, Z. et al. Data-Driven Deep Learning for The Multi-Hump Solitons and Parameters Discovery in NLS Equations with Generalized \({\mathcal{PT}\mathcal{}}\)-Scarf-II Potentials. Neural Process Lett 55, 2687–2705 (2023). https://doi.org/10.1007/s11063-022-10979-3

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