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

arXiv: https://arxiv.org/abs/2509.18479 · HTML: https://mlsanity.com/paper/2509.18479/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 14 of its 53 references that are in the map.
- First posted on arXiv 23 Sep 2025.

## Builds on (most cited first, 14 of 14)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Grad-CAM: Visual Explanations from Deep Networks via Gradient-based Localization](https://mlsanity.com/paper/1610.02391/) (2016)
- [A ConvNet for the 2020s](https://mlsanity.com/paper/2201.03545/) (2022)
- [Complexity of Linear Regions in Deep Networks](https://mlsanity.com/paper/1901.09021/) (2019)
- [Deep ReLU Networks Have Surprisingly Few Activation Patterns](https://mlsanity.com/paper/1906.00904/) (2019)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Deep Learning using Rectified Linear Units (ReLU)](https://mlsanity.com/paper/1803.08375/) (2018)
- [Impact of Fully Connected Layers on Performance of Convolutional Neural Networks for Image Classification](https://mlsanity.com/paper/1902.02771/) (2019)
- [Parameter Estimation with Dense and Convolutional Neural Networks Applied to the FitzHugh-Nagumo ODE](https://mlsanity.com/paper/2012.06691/) (2020)
- [Inverse Problem of Nonlinear Schr"odinger Equation as Learning of Convolutional Neural Network](https://mlsanity.com/paper/2107.08593/) (2021)

## Abstract (arXiv)

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 n2, the saturation intensity Isat, 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.