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非线性薛定谔方程（NLSE）在量子力学、非线性光学、等离子体物理、凝聚态物理、光纤通信和激光系统设计等多个领域中都具有重要的应用，其精确求解对于理解复杂物理现象至关重要。本文深入研究了传统的有限差分法(FDM)、分步傅里叶法(SSF)与智能算法中的物理信息神经网络(PINN)方法，旨在高效且准确地求解光纤中的复杂NLSE。这里首先介绍了PINN方法对NLSE的求解方法、步骤和结果，并对比了FDM、SSF、PINN方法对复杂NLSE求解与脉冲远距离脉冲传输的误差。然后，讨论了PINN不同网络结构和网络参数对NLSE求解精度的影响，还验证了集成学习策略的有效性，即通过结合传统数值方法与PINN的优势，提高NLSE求解的准确度。最后，采用上述算法研究了不同啁啾的艾里脉冲在光纤中的演化过程与保偏光纤对应的矢量非线性薛定谔方程求解过程。本研究通过对比FDM、SSF、PINN在求解NLSE时的特点，提出的集成学习方案在脉冲传输动力学研究和数据驱动仿真方面具有重要的应用。Nonlinear Schrodinger equation (NLSE) has important applications in quantum mechanics, nonlinear optics, plasma physics, condensed matter physics, optical fiber communication and laser system design, and its accurate solution is very important for understanding complex physical phenomena. Here, the traditional Finite Difference Method (FDM), the Split-Step Fourier Method (SSF) and the Physics-Informed Neural Network (PINN) method are studied, aiming to deeply analyze the solving mechanism of various algorithms, and then realize the efficient and accurate solution of complex nonlinear Schrodinger equation (NLSE) in optical fiber. Initially, the steps, process and results of PINN in solving the NLSE for pulse under short-distance transmission are described, and these methods’ errors are quantitatively evaluated by comparing with PINN, FDM and SSF. On this basis, the key factors affecting the accuracy of NLSE solution for pulse under long-distance transmission are further discussed. Then, the effects of different networks, activation functions, hidden layers and the number of neurons in PINN on the NLSE solution’s accuracy are discussed. It is found that selecting a suitable combination of activation functions and network types can significantly reduce the error, and the combination of FNN and tanh activation functions is particularly good. The effectiveness of Ensemble Learning strategy is also verified, that is, by combining the advantages of traditional numerical methods and PINN, the accuracy of NLSE solution is improved. Finally, the evolution characteristics of Airy pulse with different chirps in fiber and the solution of vector NLSE corresponding to polarization-maintaining fiber are studied by using the above algorithm. This study explores the solving mechanism of FDM, SSF and PINN in complex NLSE, compares and analyzes the error characteristics of those methods in various transmission scenarios, proposes and verifies the Ensemble Learning strategy. It provides a solid theoretical basis for the research of pulse transmission dynamics and data-driven simulation.
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Keywords:
- Pulse transmission in fiber 1 /
- Nonlinear Schrodinger equation 2 /
- Physical information neural network 3 /
- Airy chirp pulse 4
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