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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.
}, issn = {}, doi = {https://doi.org/10.4208/aam.OA-2024-0023}, url = {http://global-sci.org/intro/article_detail/aam/23962.html} }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.