Machine learning approach to single-shot multiparameter estimation for the non-linear Schr\"odinger equation

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.