Machine learning approach to single-shot multiparameter estimation for the non-linear Schrödinger equation

2025/09/23 by Rossignol, Louis, Aladjidi, Tangui, Baker-Rasooli, Myrann +1
#Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #FOS: Physical sciences #Optics (physics.optics) #Quantum Physics (quant-ph)

paper · doi:10.48550/arxiv.2509.18479

Abstract

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 n₂, 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.

Citations

• Observation of Jones-Roberts solitons in a paraxial quantum fluid of light
• Spin and Density Modes in a Binary Fluid of Light
• A clever neural network in solving inverse problems of Schrödinger equation
• Transit effects for non-linear index measurement in hot atomic vapors
• A ConvNet for the 2020s
• A ConvNet for the 2020s
• Inverse Problem of Nonlinear Schrödinger Equation as Learning of Convolutional Neural Network
• Parameter Estimation with Dense and Convolutional Neural Networks Applied to the FitzHugh-Nagumo ODE
• An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale
• PyTorch: An Imperative Style, High-Performance Deep Learning Library
• Complexity of Linear Regions in Deep Networks
• Deep Learning using Rectified Linear Units (ReLU)
• Attention Is All You Need
• Solving ill-posed inverse problems using iterative deep neural networks
• A Study of a Nonlinear Schrödinger Equation for Optical Fibers
• Grad-CAM: Visual Explanations from Deep Networks via Gradient-Based Localization
• Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift
• Adam: A Method for Stochastic Optimization
• ImageNet classification with deep convolutional neural networks
• Improving neural networks by preventing co-adaptation of feature detectors
• Backpropagation Applied to Handwritten Zip Code Recognition

Related

• Inverse Problem of Nonlinear Schrödinger Equation as Learning of Convolutional Neural Network
• A clever neural network in solving inverse problems of Schrödinger equation
• Physics-oriented learning of nonlinear Schrödinger equation: optical fiber loss and dispersion profile identification
• Improve the Fitting Accuracy of Deep Learning for the Nonlinear Schrödinger Equation Using Linear Feature Decoupling Method
• Data-driven femtosecond optical soliton excitations and parameters discovery of the high-order NLSE using the PINN