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Data-Driven Solutions and Parameters Discovery of the Chiral Nonlinear Schrödinger Equation via Deep Learning

Zekang Wu, Lijun Zhang (), Xuwen Huo and Chaudry Masood Khalique
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Zekang Wu: College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
Lijun Zhang: School of Science, Zhejiang University of Science and Technology, Hangzhou 310023, China
Xuwen Huo: School of Computer Science and Technology, Zhejiang Sci-Tech University, Hangzhou 310018, China
Chaudry Masood Khalique: College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China

Mathematics, 2025, vol. 13, issue 15, 1-25

Abstract: The chiral nonlinear Schrödinger equation (CNLSE) serves as a simplified model for characterizing edge states in the fractional quantum Hall effect. In this paper, we leverage the generalization and parameter inversion capabilities of physics-informed neural networks (PINNs) to investigate both forward and inverse problems of 1D and 2D CNLSEs. Specifically, a hybrid optimization strategy incorporating exponential learning rate decay is proposed to reconstruct data-driven solutions, including bright soliton for the 1D case and bright, dark soliton as well as periodic solutions for the 2D case. Moreover, we conduct a comprehensive discussion on varying parameter configurations derived from the equations and their corresponding solutions to evaluate the adaptability of the PINNs framework. The effects of residual points, network architectures, and weight settings are additionally examined. For the inverse problems, the coefficients of 1D and 2D CNLSEs are successfully identified using soliton solution data, and several factors that can impact the robustness of the proposed model, such as noise interference, time range, and observation moment are explored as well. Numerical experiments highlight the remarkable efficacy of PINNs in solution reconstruction and coefficient identification while revealing that observational noise exerts a more pronounced influence on accuracy compared to boundary perturbations. Our research offers new insights into simulating dynamics and discovering parameters of nonlinear chiral systems with deep learning.

Keywords: physics-informed neural networks; chiral nonlinear Schrödinger equations; data-driven solutions; parameters discovery (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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