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Chinese Physics B

Three-dimensional ResNet for efficient prediction of ground state phases in multicomponent dipolar spinor BECs

Chunlin Liao, Tiantian Li, Xiaodong Bai, Yunbo Zhang

Published Apr 2, 2025• 0 citations
Publisher

Abstract

Abstract Machine learning has rapidly become a powerful tool for addressing challenges in ultracold atomic systems; however, its application to intricate three-dimensional (3D) systems remains relatively underexplored. In this study, we introduce a 3D residual network (3D ResNet) framework based on 3D convolutional neural networks (3D CNN) to predict ground states phases in 3D dipolar spinor Bose–Einstein condensates (BECs). Our results show that the 3D ResNet framework predicts ground states with high accuracy and efficiency across a broad parameter space. To enhance phase transition predictions, we incorporate data augmentation techniques, leading to a notable improvement in the model’s performance. The method is further validated in more complex scenarios, particularly when transverse magnetic fields are introduced. Compared to conventional imaginary-time evolution methods (ITEM), the 3D ResNet drastically reduces computational costs, offering a rapid and scalable solution for complex 3D multi-parameter nonlinear systems.

DOI
10.1088/1674-1056/adc7f9

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Abstract This paper introduces a novel numerical method based on an energy-minimizing normalized residual network (EM-NormResNet) to compute the ground-state solution of Bose–Einstein condensates at zero or low temperatures. Starting from the three-dimensional Gross–Pitaevskii equation (GPE), we reduce it to the 1D and 2D GPEs because of the radial symmetry and cylindrical symmetry. The ground-state solution is formulated by minimizing the energy functional under constraints, which is directly solved using the EM-NormResNet approach. The paper provides detailed solutions for the ground states in 1D, 2D (with radial symmetry), and 3D (with cylindrical symmetry). We use the Thomas–Fermi approximation as the target function to pre-train the neural network. Then, the formal network is trained using the energy minimization method. In contrast to traditional numerical methods, our neural network approach introduces two key innovations: (i) a novel normalization technique designed for high-dimensional systems within an energy-based loss function; (ii) improved training efficiency and model robustness by incorporating gradient stabilization techniques into residual networks. Extensive numerical experiments validate the method’s accuracy across different spatial dimensions.
In this work, a coupled theory-guided neural network (coupled-TgNN) is constructed to explore the ground states of one-dimensional binary Bose-Einstein condensates with spin-orbit coupling and a Lee-Huang-Yang term. We find that this method is markedly superior to the ordinary deep neural network due to both the theoretical guidance for the underlying problem and the coupling of neural networks. The former shows better accuracy and more strong robustness and can rapidly construct any ground state using only imaginary-time evolution data of solutions as the training data, without the tedious step-by-step iterative calculation process. In addition, based on the coupled-TgNN approach, a phase transition boundary is also discovered, which clearly distinguishes the single-peak ground state phase from the striped phase. The results not only greatly reduce computational time for exploring the properties of the ground states but also provide a promising technique for discovering phase transitions in other coupled nonlinear systems.
Machine learning has been a powerful tool to study various models in many fields, which requires plenty of data to ensure accuracy. It is a challenge to reduce the dependence on the amount of data and ensure that the predictions are both accurate and physical. We develop a theory-guided neural network (TgNN) to explore the ground states of two-component spin–orbit coupled Bose–Einstein condensation in two-dimension. Only randomly selected few data from the twelve ground states as training data, TgNN performs higher accuracy, broader generalization ability and stronger robustness than deep neural network (DNN). Moreover, in the neighborhood of the phase transition where the predictions of DNN have obvious deviations, while TgNN still keeps relative accuracy. TgNN is less dependent on data and exhibits potential for high-dimensional and complicated models. • TgNN accurately predicts all ground states of 2D SOC BEC, which uses only one-tenth of the total data. • TgNN provides higher accuracy, broader generality ability and stronger robustness than DNN. • TgNN has more advantages than DNN in solving critical problems.
In this article, we propose an efficient and spectrally accurate numerical method to compute the ground states of three-dimensional (3D) rotating dipolar Bose-Einstein condensates (BEC) under strongly anisotropic trapping potentials.The kernel singularity, convolution non-locality and density anisotropy together complicate the dipolar potential evaluation. The fast rotation mechanism not only induces a complicated energy landscape with many local minima, but also creates a large number of vortices in the condensates. Such factors collectively make the ground state computation challenging in terms of convergence, accuracy and efficiency, especially for 3D anisotropic systems. Coupled with Fourier spectral discretization, we proposed a preconditioned conjugate gradient method (PCG) by integrating the anisotropic truncated kernel method (ATKM) for the dipolar potential evaluation. An adaptive step size control strategy is designed and ATKM allows for a spectral accuracy without introducing any extra anisotropy-dependent memory requirement or computational time. Our algorithm is spectrally accurate, highly efficient and memory-economic. Extensive numerical results are presented to confirm the accuracy and efficiency, together with applications to study impacts of the model parameters on critical rotational frequency, energies and chemical potential. Furthermore, these simulations reveal additional novel ground state patterns, such as bent vortices.
Deep learning, accounting for the use of an elaborate neural network, has recently been developed as an efficient and powerful tool to solve diverse problems in physics and other sciences. In the present work, we propose a novel learning method based on a hybrid network integrating two different kinds of neural networks: Long Short-Term Memory (LSTM) and Deep Residual Network (ResNet), in order to overcome the difficulty met in numerically simulating strongly-oscillating dynamical evolutions of physical systems. By taking the dynamics of Bose-Einstein condensates in a double-well potential as an example, we show that our new method makes a highly efficient pre-learning and a high-fidelity prediction about the whole dynamics. This benefits from the advantage of the combination of the LSTM and the ResNet and is impossibly achieved with a single network in the case of direct learning. Our method can be applied for simulating complex cooperative dynamics in a system with fast multiple-frequency oscillations with the aid of auxiliary spectrum analysis.
A Bose-Fermi mixture, consisting of both bosons and fermions, exhibits distinctive quantum coherence and phase transitions, offering valuable insights into many-body quantum systems. The ground state, as the system’s lowest energy configuration, is essential for understanding its overall behavior. In this study, we introduce the Bose-Fermi Energy-based Deep Neural Network (BF-EnDNN), a novel deep learning approach designed to solve the ground-state problem of Bose-Fermi mixtures at zero temperature through energy minimization. This method incorporates three key innovations: point sampling pre-training, a Dynamic Symmetry Layer (DSL), and a Positivity Preserving Layer (PPL). These features significantly improve the network’s accuracy and stability in quantum calculations. Our numerical results show that BF-EnDNN achieves accuracy comparable to traditional finite difference methods, with effective extension to two-dimensional systems. The method demonstrates high precision across various parameters, making it a promising tool for investigating complex quantum systems.