Machine learning approach to single-shot multiparameter estimation for the non-linear Schrödinger equation
This provides an efficient method for characterizing nonlinear systems in fields like optics and Bose-Einstein condensates, though it is incremental as it applies existing machine learning techniques to a known bottleneck.
The authors tackled the challenge of estimating strongly correlated parameters of the nonlinear Schrödinger equation from single-shot measurements by training a neural network to invert the mapping, achieving a mean absolute error of 3.22% on test data.
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.