Inverse Problem of Nonlinear Schrödinger Equation as Learning of Convolutional Neural Network
This provides a framework for solving inverse problems of nonlinear PDEs with deep learning, but it appears incremental as it applies existing deep learning methods to a specific domain.
The authors tackled the inverse problem of the nonlinear Schrödinger equation, used in fiber-optic communications, by proposing an explainable convolutional neural network (NLS-Net) and achieved a relatively accurate estimate of the parameters.
In this work, we use an explainable convolutional neural network (NLS-Net) to solve an inverse problem of the nonlinear Schrödinger 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.