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.
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.
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.
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.
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.
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 the numerical results of the structure and dynamics of the self-bound ground state arising solely because of the presence of beyond-mean-field quantum fluctuation in spin-orbit- (SO-) coupled binary Bose-Einstein condensates in one dimension. We obtain an analytical soliton solution for nonzero SO coupling that matches quite well with the numerical results. Further, we investigate the dynamical stability of these solitons by adopting three protocols, such as (i) adding initial velocity to each component, (ii) quenching the SO- and Rabi-coupling parameters at initial and finite time, and (iii) allowing collision between the two spin components by giving equal and opposite direction velocity to them. For small velocity perturbation, the soliton has the presence of the breathing oscillation, while large velocity perturbations transform the soliton into moving solitons. The maximum breathing frequency exhibits power-law dependence on the Rabi-coupling frequency with an exponent $\ensuremath{\sim}0.16$. The quenching of SO and Rabi couplings results in several interesting dynamical features, such as dynamical phase transition from plane to stripe phase, dynamical flipping of the spin state, and secondary solitonic modes. In the absence of the SO and Rabi couplings, depending upon the velocity of the up- and down-spin components, the collision between them is either elastic or inelastic, which is consistent with the earlier numerical and experimental observations. However, in the presence of coupling parameters, the collision appears to be inelastic and quasielastic in nature.
We investigate the formation, stability, and dynamics of solitons in a one-dimensional binary Bose-Einstein condensate under the action of the spin-orbit-coupling and Lee-Huang-Yang (LHY) correction to the underlying system of the Gross-Pitaevskii equations. We identify the semi-dipole (SD) family of solitons and thoroughly analyze its properties. The numerical analysis reveals intricate bifurcations, including transitions from real to complex-valued stationary wave functions of the SD solitons and norm-dependent dynamical instabilities. Stability maps in the plane of the solitons' norm and interaction strength exhibit areas of monostability, oscillatory behavior, and soliton splitting. Solitons with complex stationary wave functions emerge as ground states in broad parameter areas due to the effects of the LHY terms. The other soliton species, in the form of mixed modes (MMs), do not feature the compexification bifurcation. In the LHY-dominated regime, the SD and MM solitons exhibit identical values of the energy for the same norm. The results deepen the understanding of nonlinear matter-wave states and reveal multistable ones in quantum gases.