Deep Neural Network Approaches for Computing the Defocusing Action Ground State of Nonlinear Schrödinger Equation
Year: 2025
Author: Zhipeng Chang, Zhenye Wen, Xiaofei Zhao
Annals of Applied Mathematics, Vol. 41 (2025), Iss. 1 : pp. 42–76
Abstract
The defocusing action ground state of the nonlinear Schrödinger equation can be characterized via three different but equivalent minimization formulations. In this work, we propose some deep neural network (DNN) approaches to compute the action ground state through the three formulations. We first consider the unconstrained formulation, where we propose the DNN with a shift layer and demonstrate its necessity towards finding the correct ground state. The other two formulations involve the $L^{p+1}$-normalization or the Nehari manifold constraint. We enforce them as hard constraints into the networks by further proposing a normalization layer or a projection layer to the DNN. Our DNNs can then be trained in an unconstrained and unsupervised manner. Systematical numerical experiments are conducted to demonstrate the effectiveness and superiority of the approaches.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aam.OA-2024-0023
Annals of Applied Mathematics, Vol. 41 (2025), Iss. 1 : pp. 42–76
Published online: 2025-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 35
Keywords: Nonlinear Schrödinger equation action ground state deep neural network shift layer normalization layer projection layer.