# Inverse Problem of Nonlinear Schr"odinger Equation as Learning of Convolutional Neural Network

arXiv: https://arxiv.org/abs/2107.08593 · HTML: https://mlsanity.com/paper/2107.08593/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 59 references that are in the map.
- First posted on arXiv 19 Jul 2021.

## Builds on (most cited first, 23 of 23)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Explainable Artificial Intelligence (XAI): Concepts, Taxonomies, Opportunities and Challenges toward Responsible AI](https://mlsanity.com/paper/1910.10045/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Generating Sequences With Recurrent Neural Networks](https://mlsanity.com/paper/1308.0850/) (2013)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [MoDL: Model Based Deep Learning Architecture for Inverse Problems](https://mlsanity.com/paper/1712.02862/) (2017)
- [Learning and Generalization in Overparameterized Neural Networks, Going Beyond Two Layers](https://mlsanity.com/paper/1811.04918/) (2018)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [NETT: Solving Inverse Problems with Deep Neural Networks](https://mlsanity.com/paper/1803.00092/) (2018)
- [The phase diagram of approximation rates for deep neural networks](https://mlsanity.com/paper/1906.09477/) (2019)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Learning to solve inverse problems using Wasserstein loss](https://mlsanity.com/paper/1710.10898/) (2017)
- [Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems](https://mlsanity.com/paper/2103.15138/) (2021)
- [Adversarially learned iterative reconstruction for imaging inverse problems](https://mlsanity.com/paper/2103.16151/) (2021)

## Built on by (most cited first, 1 of 1)

- [Machine learning approach to single-shot multiparameter estimation for the non-linear Schr"odinger equation](https://mlsanity.com/paper/2509.18479/) (2025)

## Abstract (arXiv)

In this work, we use an explainable convolutional neural network (NLS-Net) to solve an inverse problem of the nonlinear Schr"odinger equation, which is widely used in fiber-optic communications. The landscape and minimizers of the non-convex loss function of the learning problem are studied empirically. It provides a guidance for choosing hyper-parameters of the method. The estimation error of the optimal solution is discussed in terms of expressive power of the NLS-Net and data. Besides, we compare the performance of several training algorithms that are popular in deep learning. It is shown that one can obtain a relatively accurate estimate of the considered parameters using the proposed method. The study provides a natural framework of solving inverse problems of nonlinear partial differential equations with deep learning.