|
| Deep Neural Network Approaches forComputing the Defocusing Action GroundState of Nonlinear Schr¨odinger Equation |
| Zhipeng Chang,Xiaofei Zhao |
| (School of Mathematics and Statistics, Wuhan University, Wuhan,
Hubei 430072, China;Computational Sciences Hubei Key Laboratory, Wuhan University, Wuhan,
Hubei 430072, China) |
| DOI： |
| Abstract: |
| The defocusing action ground state of the nonlinear Schr¨odinger
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. |
| Key words: Nonlinear Schr¨odinger equation, action ground state, deep neural network,
shift layer, normalization layer, projection layer. |