@Article{AAM-41-42,
author = {Chang , ZhipengWen , Zhenye and Zhao , Xiaofei},
title = {Deep Neural Network Approaches for Computing the Defocusing Action Ground State of Nonlinear Schrödinger Equation},
journal = {Annals of Applied Mathematics},
year = {2025},
volume = {41},
number = {1},
pages = {42--76},
abstract = {<p style="text-align: justify;">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.</p>},
issn = {},
doi = {https://doi.org/10.4208/aam.OA-2024-0023},
url = {http://global-sci.org/intro/article_detail/aam/23962.html}
}