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

Springer Science and Business Media LLC - Tập 55 - Trang 2687-2705 - 2022
Ming Zhong1,2, Jian-Guo Zhang3, Zijian Zhou1,2, Shou-Fu Tian3, Zhenya Yan1,2
1KLMM, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China
2School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China
3School of Mathematics, China University of Mining and Technology, Xuzhou, China

Tóm tắt

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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