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Results in Physics

Theory-guided neural network for studying the ground state of 2D spin-orbit coupled Bose–Einstein condensates

Junyan Kuang, Xiaodong Bai, Wenjuan Du, Tiantian Li

Published Aug 23, 2024• 1 citation
Publisher

Abstract

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.

DOI
10.1016/j.rinp.2024.107935

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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.
The ground state of spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) is an intriguing subject that has not yet been fully explored. In particular, how to quickly obtain ground states for discovering their new structures is currently under active debate. In this work, we construct a theory-guided neural network (TgNN) to explore the ground states of one-dimensional BECs with and without SOC. We find that such method is markedly superior to the ordinary deep neural network due to theory guidance of the underlying problems. The former can replace the traditional method to directly give any ground state within a valid parameter plane using only imaginary-time evolution data of nine solutions as the training data, without the tedious step-by-step iterative calculation process. Furthermore, we show that the method of TgNN exhibits higher reliability and a better ability to generalize beyond regimes covered with the training data. This method provides a promising technique for discovering novel structures of ground states in parameterized spatiotemporal systems.
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.
An important and incompletely answered question is whether machine learning methods can be used to discover the excitation of rogue waves (RWs) in nonlinear systems, especially their dynamic properties and phase transitions. In this work, a theory-guided neural network (TgNN) is constructed to explore the RWs of one-dimensional Bose-Einstein condensates. We find that such method is superior to the ordinary deep neural network due to theory guidance of underlying problems. The former can directly give any excited location, timing, and structure of RWs using only a small amount of dynamic evolution data as the training data, without the tedious step-by-step iterative calculation process. In addition, based on the TgNN approach, a phase transition boundary is also discovered, which clearly distinguishes the first-order RW phase from the non-RW phase. The results not only greatly reduce computational time for exploring RWs, but also provide a promising technique for discovering phase transitions in parameterized nonlinear systems.
Abstract We show that both single-component and two-component Bose-Einstein condensates’ (BECs) ground states can be simulated by a deep convolutional neural network. We trained the neural network via inputting the parameters in the dimensionless Gross-Pitaevskii equation (GPE) and outputting the ground-state wave function. After the training, the neural network generates ground-state wave functions with high precision. We benchmark the neural network for either inputting different coupling strength in the GPE or inputting an arbitrary potential under the infinite double walls trapping potential, and it is found that the ground state wave function generated by the neural network gives the relative chemical potential error magnitude below 10−3. Furthermore, the neural network trained with random potentials shows prediction ability on other types of potentials. Therefore, the BEC ground states, which are continuous wave functions, can be represented by deep convolutional neural networks.
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.
We present a deep neural network (DNN) -based model (HubbardNet) to variationally find the ground-state and excited-state wave functions of the one-dimensional and two-dimensional Bose-Hubbard model. Using this model for a square lattice with MMM sites, we obtain the energy spectrum as an analytical function of the on-site Coulomb repulsion, UUU, and the total number of particles, NNN, from a single training. This approach bypasses the need to solve a new Hamiltonian for each different set of values (U,N)(U,N)(U,N) and generalizes well even for out-of-distribution (U,N)(U,N)(U,N). Using HubbardNet, we identify the two ground-state phases of the Bose-Hubbard model (Mott insulator and superfluid). We show that the DNN-parametrized solutions are in excellent agreement with results from the exact diagonalization of the Hamiltonian, and it outperforms exact diagonalization in terms of computational scaling. These advantages suggest that our model is promising for efficient and accurate computation of exact phase diagrams of many-body lattice Hamiltonians.