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.
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.
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.
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 nonlinear Schrödinger equation (NLSE) underpins nonlinear wave phenomena in optics, Bose-Einstein condensates, and plasma physics, but computing its excited states remains challenging due to nonlinearity-induced non-orthonormality. Traditional methods, such as imaginary time evolution, work for ground states but fail for excited states. We propose a neural network quantum state (NNQS) approach, parameterizing wavefunctions with neural networks to directly minimize the energy functional, enabling computation of both ground and excited states. By designing compact, interpretable network architectures, we obtain analytical approximation of solutions. We apply the solutions to a case of spatiotemporal chaos in the NLSE, demonstrating its capability to study complex chaotic dynamics. This work establishes NNQS as a tool for bridging machine learning and theoretical studies of chaotic wave systems.
In this Chapter, we review key theoretical and experimental advances in the study of extreme nonlinear wave events, called rogue waves (RWs), in both single-component attractively interacting and two-component repulsive mixtures of ultracold quantum gases. Starting from the exact rational solutions of the integrable focusing nonlinear Schroedinger model, the hierarchy of RW solutions is exemplified. These range from the Peregrine soliton (PS) and, related to it, the destabilization into a multi-peak cascade of PSs dubbed "Christmas-tree", to the Akhmediev breather, and Kuznetsov-Ma soliton as well as higher-order RWs. Emphasis is placed on their controllable dynamical emergence and characteristics in non-integrable quantum many-body systems described by Gross-Pitaevskii models and extensions thereof through different protocols such as modulational instability, gradient catastrophe, and dam-break flows. We further discuss how immiscible particle-imbalanced repulsive mixtures can be cast into effective attractive single-component environments capable of hosting RWs. Next, state-of-the-art experimental techniques are summarized within the ultracold realm that can be utilized to realize solitary waves, modulational instability, dispersive shock waves and RWs including the very recent first experimental observation of the PS, enabled through engineered effective focusing interactions and precise dynamical triggering. Observations of these extreme events in water waves, nonlinear optics and beyond are also outlined, highlighting their broader relevance and potential of emergence in disparate physical settings. Our exposition aims at showcasing ultracold atomic gases as versatile platforms for controllably generating and probing extreme nonlinear events, among others, in the quantum realm across integrable and non-integrable settings.
Identifying phase transitions and classifying phases of matter is central to understanding the properties and behavior of a broad range of material systems. In recent years, machine-learning (ML) techniques have been successfully applied to perform such tasks in a data-driven manner. However, the success of this approach notwithstanding, we still lack a clear understanding of ML methods for detecting phase transitions, particularly of those that utilize neural networks (NNs). In this work, we derive analytical expressions for the optimal output of three widely used NN-based methods for detecting phase transitions. These optimal predictions correspond to the results obtained in the limit of high model capacity. Therefore, in practice they can, for example, be recovered using sufficiently large, well-trained NNs. The inner workings of the considered methods are revealed through the explicit dependence of the optimal output on the input data. By evaluating the analytical expressions, we can identify phase transitions directly from experimentally accessible data without training NNs, which makes this procedure favorable in terms of computation time. Our theoretical results are supported by extensive numerical simulations covering, e.g., topological, quantum, and many-body localization phase transitions. We expect similar analyses to provide a deeper understanding of other classification tasks in condensed matter physics.