Computing ground states of Bose-Einstein condensation by normalized deep neural network

TL;DR

This paper introduces a normalized deep neural network (norm-DNN) that efficiently computes ground and excited states of Bose-Einstein condensation by incorporating normalization and shift layers to handle unitary constraints.

Contribution

The paper presents a novel norm-DNN architecture with specialized layers for solving constrained minimization problems in BEC, extending its application to high dimensions and excited states.

Findings

Effective in 1D, 2D, and multi-component BEC ground state computations.

Outperforms existing machine learning methods in accuracy and efficiency.

Successfully extends to excited state calculations.

Abstract

We propose a normalized deep neural network (norm-DNN) for computing ground states of Bose-Einstein condensation (BEC) via the minimization of the Gross-Pitaevskii energy functional under unitary mass normalization. Compared with the traditional deep neural network for solving partial differential equations, two additional layers are added in training our norm-DNN for solving this kind of unitary constraint minimization problems: (i) a normalization layer is introduced to enforce the unitary mass normalization, and (ii) a shift layer is added to guide the training to non-negative ground state. The proposed norm-DNN gives rise to an efficient unsupervised approach for learning ground states of BEC. Systematical investigations are first carried out through extensive numerical experiments for computing ground states of BEC in one dimension. Extensions to high dimensions and multi-component are then studied in details. The results demonstrate the effectiveness and efficiency of norm-DNN for learning ground states of BEC. Finally, we extend the norm-DNN for computing the first excited states of BEC and discuss parameter generalization issues as well as compare with some existing machine learning methods for computing ground states of BEC in the literature.