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

TL;DR

This paper introduces an explainable convolutional neural network approach to solve inverse problems of the nonlinear Schrödinger equation, providing insights into loss landscape, hyper-parameter selection, and estimation accuracy.

Contribution

It presents a novel deep learning framework (NLS-Net) for inverse PDE problems, analyzing loss landscape, hyper-parameters, and comparing training algorithms.

Findings

Relatively accurate parameter estimation achieved.

Loss landscape and minimizers characterized empirically.

Training algorithm performance compared.

Abstract

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.