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arXiv:1712.10093 · Scientific Reports

Generation of Bose-Einstein Condensates’ Ground State Through Machine Learning

Xiao Liang, Huan Zhang, Sheng Liu, Yan Li, Yong-Sheng Zhang

Published Oct 30, 2018• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/s41598-018-34725-9

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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.
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.
Machine Learning methods are emerging as faster and efficient alternatives to numerical simulation techniques. The field of Scientific Computing has started adopting these data-driven approaches to faithfully model physical phenomena using scattered, noisy observations from coarse-grained grid-based simulations. In this paper, we investigate data-driven modelling of Bose-Einstein Condensates (BECs). In particular, we use Gaussian Processes (GPs) to model the ground state wave function of BECs as a function of scattering parameters from the dimensionless Gross Pitaveskii Equation (GPE). Experimental results illustrate the ability of GPs to accurately reproduce ground state wave functions using a limited number of data points from simulations. Consistent performance across different configurations of BECs, namely Scalar and Vectorial BECs generated under different potentials, including harmonic, double well and optical lattice potentials pronounces the versatility of our method. Comparison with existing data-driven models indicates that our model achieves similar accuracy with only a small fraction ([Formula: see text]th) of data points used by existing methods, in addition to modelling uncertainty from data. When used as a simulator post-training, our model generates ground state wave functions [Formula: see text] faster than Trotter Suzuki, a numerical approximation technique that uses Imaginary time evolution. Our method is quite general; with minor changes it can be applied to similar quantum many-body problems.
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.
We identify all possible classes of solutions for two-component Bose-Einstein condensates (BECs) within the Thomas-Fermi (TF) approximation and check these results against numerical simulations of the coupled Gross-Pitaevskii equations (GPEs). We find that they can be divided into two general categories. The first class contains solutions with a region of overlap between the components. The other class consists of non-overlapping wavefunctions and also contains solutions that do not possess the symmetry of the trap. The chemical potential and average energy can be found for both classes within the TF approximation by solving a set of coupled algebraic equations representing the normalization conditions for each component. A ground state minimizing the energy (within both classes of states) is found for a given set of parameters characterizing the scattering length and confining potential. In the TF approximation, the ground state always shares the symmetry of the trap. However, a full numerical solution of the coupled GPEs, incorporating the kinetic energy of the BEC atoms, can sometimes select a broken-symmetry state as the ground state of the system. We also investigate effects of finite-range interactions on the structure of the ground state.
Abstract We propose an artificial neural network (ANN) design to solve the inverse problem for a 1D Gross–Pitaevskii equation (GPE). More precise, the ANN takes the squared modulus of the stationary GPE solution as an input and returns the parameters of the potential function and the factor in front of the GPE non-linear term. From the physical point of view the ANN predicts the parameters of a trap potential and the interaction constant of 1D Bose–Einstein condensate by its density distribution. Using the results of numerical solution of GPE for more than <?CDATA 30 00030\,00030000?> 30 000 sets of GPE parameters as train and validation datasets we build the ANN as a fast and accurate inverse GPE solver.
In this paper, we assess whether Physics-Informed Neural Networks (PINNs) can capture spontaneous symmetry breaking (SSB) in the stationary Gross–Pitaevskii equation for double-well potentials. For attractive interactions ([Formula: see text]), the PINN successfully captures the qualitative features of macroscopic quantum self-trapping, yielding a robust asymmetric population imbalance ([Formula: see text]). For repulsive interactions ([Formula: see text]), the unregularized network converges to spurious asymmetric solutions, but a continuous parity regularization term effectively restores the symmetric ground state ([Formula: see text]). Benchmarking against a spectral imaginary-time reference, the post-hoc chemical potential agrees to within 2.3% across the full interaction range and to better than 0.6% for [Formula: see text], with density errors of order [Formula: see text]–[Formula: see text]. We reproduce the two-mode model parameters ([Formula: see text], B, [Formula: see text]) using full GPE eigenstates and show that the asymmetric branch of the two-mode model, although formally present ([Formula: see text]), corresponds to higher-energy self-trapped states for repulsive interactions, explaining why the ground state remains symmetric for [Formula: see text].