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Toward fast, accurate and robust AI prediction of ground states in rotating BEC

Zhizhong Kong, Jerry Zhijian Yang, Cheng Yuan, Xiaofei Zhao

TL;DR

This work tackles computing the ground state of rotating Bose-Einstein condensates by solving a constrained nonlinear eigenproblem for the rotating GPE. It introduces a mass-preserving normalized loss together with a training strategy called virtual rotation acceleration to reliably capture vortex-rich ground states across diverse rotation rates and confinements; it further develops adaptive sampling and an unsupervised operator-learning framework to generalize GS predictions over physical parameters via distillation into a unified model. The results show high accuracy across 2D and 3D regimes and demonstrate robustness to phase transitions, enabling rapid GS predictions and useful inverse problems. The approach offers a scalable, unsupervised alternative to traditional gradient-flow methods, with potential for fast online GS estimation and parameter inference in rotating quantum fluids.

Abstract

We propose an unsupervised deep learning approach for computing the ground state (GS) of rotating Bose-Einstein condensation. To minimize the energy under a mass constraint, our approach introduces two key and novel ingredients: a normalized loss function that exactly enforces the mass constraint, and a training strategy named virtual rotation acceleration that is essential for avoiding local minima and guiding the learning process to the correct quantized vortex phase. Extensive numerical experiments demonstrate the proposed approach as an effective and accurate method to predict GS across physical conditions--from slow to fast rotation and from isotropic to anisotropic confinement. Through further distillation, we establish a unified operator network capable of efficiently generalizing physical parameters across different phases. It enables rapid GS predictions while correctly capturing phase transitions and is applied for inverse problems.

Toward fast, accurate and robust AI prediction of ground states in rotating BEC

TL;DR

This work tackles computing the ground state of rotating Bose-Einstein condensates by solving a constrained nonlinear eigenproblem for the rotating GPE. It introduces a mass-preserving normalized loss together with a training strategy called virtual rotation acceleration to reliably capture vortex-rich ground states across diverse rotation rates and confinements; it further develops adaptive sampling and an unsupervised operator-learning framework to generalize GS predictions over physical parameters via distillation into a unified model. The results show high accuracy across 2D and 3D regimes and demonstrate robustness to phase transitions, enabling rapid GS predictions and useful inverse problems. The approach offers a scalable, unsupervised alternative to traditional gradient-flow methods, with potential for fast online GS estimation and parameter inference in rotating quantum fluids.

Abstract

We propose an unsupervised deep learning approach for computing the ground state (GS) of rotating Bose-Einstein condensation. To minimize the energy under a mass constraint, our approach introduces two key and novel ingredients: a normalized loss function that exactly enforces the mass constraint, and a training strategy named virtual rotation acceleration that is essential for avoiding local minima and guiding the learning process to the correct quantized vortex phase. Extensive numerical experiments demonstrate the proposed approach as an effective and accurate method to predict GS across physical conditions--from slow to fast rotation and from isotropic to anisotropic confinement. Through further distillation, we establish a unified operator network capable of efficiently generalizing physical parameters across different phases. It enables rapid GS predictions while correctly capturing phase transitions and is applied for inverse problems.

Paper Structure

This paper contains 17 sections, 1 theorem, 32 equations, 14 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Capturing GS with vortices
  3. Preliminaries
  4. DNN with normalized loss
  5. Virtual rotation acceleration (VRA): a key training strategy
  6. Numerical studies
  7. Necessity of VRA
  8. More validation of the method
  9. Normalized loss vs normalized network
  10. Adaptive sampling
  11. Generalization in physical parameters
  12. Piecewise trained DNN
  13. A unified model with distillation
  14. Distillation
  15. Distilled model as foundation model
  16. ...and 2 more sections

Key Result

Theorem 3.8

Consider the 2D GS problem with a fixed $\Omega$ and varying $\gamma$. Let $D \subset \mathbb{R}_{+} \cup {0}$ be an interval on which $\mathcal{E}(\gamma)$ is defined. Then, $\mathcal{E}(\gamma)$ is strictly monotonically increasing and concave over $D$.

Figures (14)

  • Figure 1: Total number of vortices and the energy of GS under different $\Omega$.
  • Figure 1: Surface plot of $|\psi_\theta|^2$ for different $\Omega\in U$ in Example \ref{['omega_omega']} with $M=2$.
  • Figure 2: Surface plot of $\vert\psi_{\theta}\vert^2$ under $\Omega = 0.5$ and $\beta=100$ at different training epochs without VRA (1st row) or with VRA (2nd row).
  • Figure 2: Surface plot of $|\psi_\theta|^2$ for different $\gamma_y\in U$ in Example \ref{['gamma_gamma']} with $M=2$.
  • Figure 3: Energy of numerical solution obtained from training without VRA (1st row) and with VRA (2nd row) for different values of $\Omega$ under $\beta=100$.
  • ...and 9 more figures

Theorems & Definitions (15)

  • Remark 2.1
  • Example 2.2: GS in different rotating regimes
  • Example 2.3: GS under different confinement ratio
  • Example 2.4: Three-dimensional case 3Dexample
  • Example 3.1: Generalizing $\Omega$ in one phase
  • Example 3.2: Generalizing $\gamma_y$ in one phase
  • Remark 3.3
  • Example 3.4: Failure of generalization across phases
  • Remark 3.5
  • Example 3.6: Generalizing $\Omega$ in multi-phase
  • ...and 5 more