用神经网络从单次测量中快速估出非线性薛定谔方程的三个关键参数。
Machine learning approach to single-shot multiparameter estimation for the non-linear Schrödinger equation
- 将参数估计建模为逆问题，用ConvNeXt神经网络学习非线性薛定谔方程的反映射。
- 在12500个未见样本上实现3.22%的平均绝对误差，性能稳定且泛化能力强。
- 适用于光学、玻色-爱因斯坦凝聚等非线性系统快速表征，适合实验与仿真结合场景。
非线性薛定谔方程（NLSE）是描述光纤到玻色-爱因斯坦凝聚体等多种非线性介质中波动力学的基础模型。从单次测量中准确估计其参数（常高度相关）仍是重大挑战。本文将参数估计视为逆问题，训练神经网络以反演NLSE映射。结合快速数值求解器与基于ConvNeXt架构及多变量高斯负对数似然损失函数的机器学习方法，模型仅凭单次场（密度与相位）图像即可同时估计三个关键参数：非线性系数 $n_2$、饱和强度 $I_{sat}$ 与线性吸收系数 $α$。模型在10万张模拟图像上训练，在12500个未见测试样本上达到3.22%的平均绝对误差，展现出优异泛化能力并接近真实值。该方法为非线性系统的高效表征提供了新路径，具备在引入真实噪声后连接理论建模与实验数据的潜力。
原文摘要 · Abstract (English)
The nonlinear Schrödinger equation (NLSE) is a fundamental model for wave dynamics in nonlinear media ranging from optical fibers to Bose-Einstein condensates. Accurately estimating its parameters, which are often strongly correlated, from a single measurement remains a significant challenge. We address this problem by treating parameter estimation as an inverse problem and training a neural network to invert the NLSE mapping. We combine a fast numerical solver with a machine learning approach based on the ConvNeXt architecture and a multivariate Gaussian negative log-likelihood loss function. From single-shot field (density and phase) images, our model estimates three key parameters: the nonlinear coefficient $n_2$, the saturation intensity $I_{sat}$, and the linear absorption coefficient $α$. Trained on 100,000 simulated images, the model achieves a mean absolute error of $3.22\%$ on 12,500 unseen test samples, demonstrating strong generalization and close agreement with ground-truth values. This approach provides an efficient route for characterizing nonlinear systems and has the potential to bridge theoretical modeling and experimental data when realistic noise is incorporated.
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