Data-Driven Solutions and Parameters Discovery of the Chiral Nonlinear Schrödinger Equation via Deep Learning
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 forw...
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2025-07-01
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| Series: | Mathematics |
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| author | Zekang Wu Lijun Zhang Xuwen Huo Chaudry Masood Khalique |
| author_facet | Zekang Wu Lijun Zhang Xuwen Huo Chaudry Masood Khalique |
| author_sort | Zekang Wu |
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| description | 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. |
| format | Article |
| id | doaj-art-746b8a10cb4541e992ab5739bb923697 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
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| series | Mathematics |
| spelling | doaj-art-746b8a10cb4541e992ab5739bb9236972025-08-20T03:36:41ZengMDPI AGMathematics2227-73902025-07-011315234410.3390/math13152344Data-Driven Solutions and Parameters Discovery of the Chiral Nonlinear Schrödinger Equation via Deep LearningZekang Wu0Lijun Zhang1Xuwen Huo2Chaudry Masood Khalique3College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, ChinaSchool of Science, Zhejiang University of Science and Technology, Hangzhou 310023, ChinaSchool of Computer Science and Technology, Zhejiang Sci-Tech University, Hangzhou 310018, ChinaCollege of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, ChinaThe 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.https://www.mdpi.com/2227-7390/13/15/2344physics-informed neural networkschiral nonlinear Schrödinger equationsdata-driven solutionsparameters discovery |
| spellingShingle | Zekang Wu Lijun Zhang Xuwen Huo Chaudry Masood Khalique Data-Driven Solutions and Parameters Discovery of the Chiral Nonlinear Schrödinger Equation via Deep Learning Mathematics physics-informed neural networks chiral nonlinear Schrödinger equations data-driven solutions parameters discovery |
| title | Data-Driven Solutions and Parameters Discovery of the Chiral Nonlinear Schrödinger Equation via Deep Learning |
| title_full | Data-Driven Solutions and Parameters Discovery of the Chiral Nonlinear Schrödinger Equation via Deep Learning |
| title_fullStr | Data-Driven Solutions and Parameters Discovery of the Chiral Nonlinear Schrödinger Equation via Deep Learning |
| title_full_unstemmed | Data-Driven Solutions and Parameters Discovery of the Chiral Nonlinear Schrödinger Equation via Deep Learning |
| title_short | Data-Driven Solutions and Parameters Discovery of the Chiral Nonlinear Schrödinger Equation via Deep Learning |
| title_sort | data driven solutions and parameters discovery of the chiral nonlinear schrodinger equation via deep learning |
| topic | physics-informed neural networks chiral nonlinear Schrödinger equations data-driven solutions parameters discovery |
| url | https://www.mdpi.com/2227-7390/13/15/2344 |
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