Title: Machine learning approach to single-shot multiparameter estimation for the non-linear Schrödinger equation Show Chinese title

Title: 基于机器学习的非线性薛定谔方程单次多参数估计方法

Abstract: The nonlinear Schr\"odinger 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 $\alpha$. 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. Show Chinese abstract

Abstract: 非线性薛定谔方程（NLSE）是研究从光纤到玻色-爱因斯坦凝聚体的非线性介质中波动力学的基本模型。 从单次测量中准确估计其参数（这些参数通常高度相关）仍然是一个重大挑战。 我们通过将参数估计视为一个逆问题，并训练一个神经网络来反转NLSE映射来解决这个问题。 我们将一个快速数值求解器与基于ConvNeXt架构和多变量高斯负对数似然损失函数的机器学习方法相结合。 从单次拍摄的场（密度和相位）图像中，我们的模型估计了三个关键参数：非线性系数$n_2$，饱和强度$I_{sat}$和线性吸收系数$\alpha$。 在100,000个模拟图像上进行训练后，该模型在12,500个未见过的测试样本上的平均绝对误差为$3.22\%$，表现出强大的泛化能力，并与真实值高度一致。 这种方法为表征非线性系统提供了一种高效途径，并在引入现实噪声时有望弥合理论建模与实验数据之间的差距。