High-dimensional Systems Research Articles

Intermittent phase dynamics of non-autonomous oscillators through time-varying phase

Oscillatory dynamics pervades the universe, appearing in systems of all scales. Whilst autonomous oscillatory dynamics has been extensively studied and is well understood, the very important problem of non-autonomous oscillatory dynamics is less well understood. Here, we provide a framework for non-autonomous oscillatory dynamics, within which we can define intermittent phenomena such as intermittent phase synchronisation. Moreover, we demonstrate this framework with a coupled pair of non-autonomous phase oscillators as well as a higher-dimensional system comprising of two interacting phase-oscillator networks.

Gaussian Process Emulation for High-Dimensional Coupled Systems

Complex coupled multiphysics simulations are ubiquitous in science and engineering. Evaluating these numerical simulators is often costly which limits our ability to run them sufficiently often for forward uncertainty quantification. Furthermore outputs are generally not scalar quantities but depend on space and/or time. Gaussian process emulators are statistical surrogates which can approximate the output of the complex computer models at untested inputs quickly while also providing uncertainty information about the accuracy of evaluating the emulator rather than the full physical model. GP emulators were originally developed in the context of scalar output from a single physical model but have since been extended to vector-valued quantities of interest (parallel partial emulators) and to coupled physics by connecting two independent emulators, one for each type of physics (linked emulation). The parallel partial linked GP emulator (PPLE) developed in this work combines the efficiency of a shared correlation structure with the accuracy of linked emulators to produce a new tool for emulating compositions of functions with vector-valued output. The PPLE applied to two numerical experiments out-performs direct emulation of the output composite function producing results with smaller average prediction error and variance.

Hierarchical generalization of dual unitarity

Quantum dynamics with local interactions in lattice models display rich physics, but is notoriously hard to study. Dual-unitary circuits allow for exact answers to interesting physical questions in clean or disordered one- and higher-dimensional quantum systems. However, this family of models shows some non-universal features, like vanishing correlations inside the light-cone and instantaneous thermalization of local observables. In this work we propose a generalization of dual-unitary circuits where the exactly calculable spatial-temporal correlation functions display richer behavior, and have non-trivial thermalization of local observables. This is achieved by generalizing the single-gate condition to a hierarchy of multi-gate conditions, where the first level recovers dual-unitary models, and the second level exhibits these new interesting features. We also extend the discussion and provide exact solutions to correlators with few-site observables and discuss higher-orders, including the ones after a quantum quench. In addition, we provide exhaustive parametrizations for qubit cases, and propose a new family of models for local dimensions larger than two, which also provides a new family of dual-unitary models.

EPR-Net: constructing a non-equilibrium potential landscape via a variational force projection formulation.

We present EPR-Net, a novel and effective deep learning approach that tackles a crucial challenge in biophysics: constructing potential landscapes for high-dimensional non-equilibrium steady-state systems. EPR-Net leverages a nice mathematical fact that the desired negative potential gradient is simply the orthogonal projection of the driving force of the underlying dynamics in a weighted inner-product space. Remarkably, our loss function has an intimate connection with the steady entropy production rate (EPR), enabling simultaneous landscape construction and EPR estimation. We introduce an enhanced learning strategy for systems with small noise, and extend our framework to include dimensionality reduction and the state-dependent diffusion coefficient case in a unified fashion. Comparative evaluations on benchmark problems demonstrate the superior accuracy, effectiveness and robustness of EPR-Net compared to existing methods. We apply our approach to challenging biophysical problems, such as an eight-dimensional (8D) limit cycle and a 52D multi-stability problem, which provide accurate solutions and interesting insights on constructed landscapes. With its versatility and power, EPR-Net offers a promising solution for diverse landscape construction problems in biophysics.

The chaotic properties and circuit design of a generalized high-dimensional integer-domain system

The purpose of this study is to formulate an innovative approach for creating a higher-dimensional digital chaotic system (HDDCS) within the realm of finite precision. The suggested methodology revolves around generating random sequences to regulate the emergence of chaotic behavior. Rigorous mathematical proofs of the chaotic characteristics of the HDDCS are given, and the nonlinear characteristics of HDIDS are demonstrated by examples. Furthermore, an analog–digital hybrid circuit is developed to design the generalized HDIDS. This study offers substantial theoretical support for the chaotic nature of the HDIDS, and validates the feasibility of chaotic circuit design by simulation.

Review and analysis of model order reduction techniques for high-dimensional complex systems

Review and analysis of model order reduction techniques for high-dimensional complex systems

A hybrid forecasting framework based on MCS and machine learning for higher dimensional and unbalanced systems

Forecasting methods and theories have been widely researched and applied in complex systems and fields such as statistical physics, econophysics, material crystals, etc. However, challenges persist in applying these methods to complex systems characterized by high dimensionality, data imbalance, and single prediction evaluation. To address these issues, we propose a novel hybrid forecasting approach that integrates the model confidence set (MCS) with machine learning (ML) models. We introduce Principal Component Analysis (PCA) to reduce dimensionality of the data, reduce data imbalance through a combination of random undersampling and oversampling, and introduce several metrics to evaluate the machine learning model set. We also introduce the MCS to select the optimal model from the set of ML models and propose a new combinatorial approach, the MCS-ML combinatorial model. An empirical study is conducted using the example of abnormal transactions in the Bitcoin blockchain. The empirical results show that the proposed MCS-ML combinatorial model has better predictive performance than the models in the ML model set under different data structures.

Quantum Steering with Imprecise Measurements.

We study quantum steering experiments without assuming that the trusted party can perfectly control their measurement device. Instead, we introduce a scenario in which these measurements are subject to small imprecision. We show that small measurement imprecision can have a large detrimental influence in terms of false positives for steering inequalities, and that this effect can become even more relevant for high-dimensional systems. We then introduce a method for taking generic measurement imprecision into account in tests of bipartite steering inequalities. The revised steering bounds returned by this method are analytical, easily computable, and are even optimal for well-known families of arbitrary-dimensional steering tests. Furthermore, it applies equally well to generalized quantum steering scenarios, where the shared quantum state does not need to be separable, but is instead limited by some other entanglement property.

Data-driven model selections of second-order particle dynamics via integrating Gaussian processes with low-dimensional interacting structures

In this paper, we focus on the data-driven discovery of a general second-order particle-based model that contains many state-of-the-art models for modeling the aggregation and collective behavior of interacting agents of similar size and body type. This model takes the form of a high-dimensional system of ordinary differential equations parameterized by two interaction kernels that appraise the alignment of positions and velocities. We propose a Gaussian Process-based approach to this problem, where the unknown model parameters are marginalized by using two independent Gaussian Process (GP) priors on latent interaction kernels constrained to dynamics and observational data. This results in a nonparametric model for interacting dynamical systems that accounts for uncertainty quantification. We also show that our estimators can be efficiently derived through the use of preconditioned techniques, ensuring scalability. Moreover, we perform a theoretical analysis to interpret the methodology and investigate the conditions under which the kernels can be recovered. We demonstrate the effectiveness of the proposed approach on various prototype systems, including the selection of the order of the systems and the types of interactions. In particular, we present applications to modeling two real-world fish motion datasets that display flocking and milling patterns up to 248 dimensions. Despite the use of small datasets, the GP-based approach learns an effective representation of the nonlinear dynamics in these spaces and outperforms competitor methods.

Hybrid synchronisation method based on inverse generalised and inverse projected high dimensional discrete chaotic systems

High-dimensional discrete chaotic systems have a wide range of engineering applications, while the chaotic synchronization method is the key to confidential communication applications. Based on the proposed discriminant theorem for high-order polynomial chaotic mapping, in this paper, a hybrid inverse generalization and inverse projection synchronization method for high-dimensional discrete chaotic systems is constructed. The method increases the flexibility of synchronization control by designing to enable the coexistence of inverse projection synchronization and inverse generalized synchronization in chaotic systems, by specifically using an invertible and adjustable constant diagonal matrix to extend the control capability of the error system. Meanwhile, this paper proposed the validity of this hybrid synchronization scheme based on the multi-stationary chaotic system for the first time, through comparison, it is certified that the multi-stationary discrete chaotic synchronization system introduced in this paper not only has complex dynamics behavior but also has a faster synchronization speed. Meanwhile, this paper proposed the validity of this hybrid synchronization scheme based on the multi-stationary chaotic system for the first time, through comparison, it is certified that the multi-stationary discrete chaotic synchronization system introduced in this paper not only has complex dynamics behavior but also has a faster synchronization speed. Finally, in this paper, the hybrid synchronization-based encryption system is also constructed, in which the transmitter switches the chaotic system between different attractors by changing the initial conditions, and then uses different chaotic attractors to mask the plaintext information. The experimental results show that the system has higher security and larger key space.

Extended Dynamic Mode Decomposition with Invertible Dictionary Learning

The Koopman operator has received attention for providing a potentially global linearization representation of the nonlinear dynamical system. To estimate or control the original system, the invertibility problem is introduced into the data-driven modeling, i.e., the observables are required to be reconstructed the original system’s states. Existing methods cannot solve this problem perfectly. Only linear or nonlinear but lossy reconstruction can be achieved. This paper proposed a novel data-driven modeling approach, denoted as the Extended Dynamic Mode Decomposition with Invertible Dictionary Learning (EDMD-IDL) to address this issue, which can be interpreted as a further extension of the classical Extended Dynamic Mode Decomposition (EDMD). The Invertible Neural Network (INN) is introduced in the proposed method, where its inverse process provides the explicit inverse on the dictionary functions, thus allowing the nonlinear and lossless reconstruction. An iterative algorithm is designed to solve the extended optimization problem defined by the Koopman operator and INN by combining the optimization algorithm based on the gradient descent and the classical EDMD method, making the method successfully obtain the finite-dimensional approximation of the Koopman operator. The method is tested on various canonical nonlinear dynamical systems and is shown that the predictions obtained in a linear fashion and the ground truth match well over the long-term, where only the initial status is provided. Comparison experiments highlight the superiority of the proposed method over the other EDMD-based methods. Notably, a typical example in fluid dynamics, cylinder wake, illustrates the potential of the method to be further extended to the high-dimensional system with tens of thousands of states. By combining the Proper Orthogonal Decomposition technique, nontrivial Kármán vortex sheet phenomenon is perfectly reconstructed. Our proposed method provides a new paradigm for solving the finite-dimensional approximation of the Koopman operator and applying it to data-driven modeling.

Single-Shot Readout of a Solid-State Electron Spin Qutrit.

Quantum systems usually feature a rich multilevel structure with promising resources for developing superior quantum technologies compared with their binary counterpart. Single-shot readout of these high-dimensional quantum systems is essential for exploiting their potential. Although there have been various high-spin systems, the single-shot readout of the overall state of high-spin systems remains a challenging issue. Here we demonstrate a reliable single-shot readout of spin qutrit state in a low-temperature solid-state system utilizing a binary readout scheme. We achieve a single-shot readout of an electron spin qutrit associated with a single nitrogen-vacancy center in diamond with an average fidelity of 87.80%. We use this spin qutrit system to verify quantum contextuality, a fundamental test of quantum mechanics. We observe a violation of the noncontextual hidden variable inequality with the developed single-shot readout in contrast to the conventional binary readout. These results pave the way for developing quantum information processing based on spin qutrits.

Learning interpretable collective variables for spreading processes on networks.

Collective variables (CVs) are low-dimensional projections of high-dimensional system states. They are used to gain insights into complex emergent dynamical behaviors of processes on networks. The relation between CVs and network measures is not well understood and its derivation typically requires detailed knowledge of both the dynamical system and the network topology. In this Letter, we present a data-driven method for algorithmically learning and understanding CVs for binary-state spreading processes on networks of arbitrary topology. We demonstrate our method using four example networks: the stochastic block model, a ring-shaped graph, a random regular graph, and a scale-free network generated by the Albert-Barabási model. Our results deliver evidence for the existence of low-dimensional CVs even in cases that are not yet understood theoretically.

Spontaneously interacting qubits from Gauss-Bonnet

Building on previous constructions examining how a collection of small, locally interacting quantum systems might emerge via spontaneous symmetry breaking from a single-particle system of high dimension, we consider a larger family of geometric loss functionals and explicitly construct several classes of critical metrics which “know about qubits” (KAQ). The loss functional consists of the Ricci scalar with the addition of the Gauss-Bonnet term, which introduces an order parameter that allows for spontaneous symmetry breaking. The appeal of this method is two-fold: (i) the Ricci scalar has already been shown to have KAQ critical metrics and (ii) exact equations of motions are known for loss functionals with generic curvature terms up to two derivatives. We show that KAQ critical metrics, which are solutions to the equations of motion in the space of left-invariant metrics with fixed determinant, exist for loss functionals that include the Gauss-Bonnet term. We find that exploiting the subalgebra structure leads us to natural classes of KAQ metrics which contain the familiar distributions (GUE, GOE, GSE) for random Hamiltonians. We introduce tools for this analysis that will allow for straightfoward, although numerically intensive, extension to other loss functionals and higher-dimension systems.

One-dimensional extended Su–Schrieffer–Heeger models as descendants of a two-dimensional topological model

The topological phase diagrams and finite-size energy spectra of one-dimensional extended Su–Schrieffer–Heeger (SSH) models with long-range hoppings on the trimer lattice are investigated in detail. Due to the long-range hoppings, the band structure of the original SSH model becomes more complicated and new phases with the large Zak phase can emerge. Furthermore, a seeming violation of bulk-edge correspondence occurs in the one-dimensional topological system whose band topology stems from the inversion symmetry. The one-dimensional models are mapped onto a two-dimensional topological model when a parameter of the one-dimensional models is regarded as an additional degree of freedom. As Fourier components of the derived two-dimensional model, phase boudaries and the finite-size spectra of one-dimensional models can be recovered from the model in the higher spatial dimensions. Then the origin of edge modes of one-dimensional models can be understood from two dimensions and we give a reasonable explanation of the violation of bulk-edge correspondence in one spatial dimension. In fact, we may give a general perspective that the topological properties of one-dimensional (lower-dimensional) systems can be found their origin from two-dimensional (higher-dimensional) systems.

Simplification of ANFIS based on importance-confidence-similarity measures

Adaptive-network-based fuzzy inference system (ANFIS) is a well-known neuro-fuzzy system, which is widely implemented in nonlinear system approximation, prediction, control, and pattern classification. However, either too many fuzzy sets or fuzzy rules are indispensable to establish an accurate ANFIS for a high-dimensional system, which increases the complexity and limits the widespread application of the ANFIS. In this paper, an approach to simplify the ANFIS is developed based on importance-confidence-similarity (ICS) measures. The simplification process consists of selection of fuzzy rules with large importance-confidence (IC) values and reduction of similar fuzzy rules. The importance is defined as the average normalized firing strength of the rule. The confidence which reflects the approximation performance of a T-S fuzzy rule is estimated by the weighted orthogonal distance from the output data to the rule hyperplane (the consequent of a linear T-S fuzzy rule). The IC measures provide a comprehensive basis for the selection of the appropriate fuzzy rules. Furthermore, to the best of our knowledge, an approach to measure similarities of the T-S fuzzy rules including the rule premises and consequents is first proposed in this paper. We also develop a direct and simple method to remove the similar fuzzy rules based on a hierarchical tree and the IC measures. The proposed simplification approach is applied to Mackey-Glass chaotic time series prediction problem and the strip temperature increment prediction problem of the indirect-fired furnace of the annealing heating line. The results show that our proposed method achieves a balance between simplicity and accuracy of the ANFIS for the high-dimensional system.

High-order geometric integrators for the local cubic variational Gaussian wavepacket dynamics.

Gaussian wavepacket dynamics has proven to be a useful semiclassical approximation for quantum simulations of high-dimensional systems with low anharmonicity. Compared to Heller's original local harmonic method, the variational Gaussian wavepacket dynamics is more accurate, but much more difficult to apply in practice because it requires evaluating the expectation values of the potential energy, gradient, and Hessian. If the variational approach is applied to the local cubic approximation of the potential, these expectation values can be evaluated analytically, but they still require the costly third derivative of the potential. To reduce the cost of the resulting local cubic variational Gaussian wavepacket dynamics, we describe efficient high-order geometric integrators, which are symplectic, time-reversible, and norm-conserving. For small time steps, they also conserve the effective energy. We demonstrate the efficiency and geometric properties of these integrators numerically on a multidimensional, nonseparable coupled Morse potential.

Pre-Training Physics-Informed Neural Network with Mixed Sampling and Its Application in High-Dimensional Systems

Pre-Training Physics-Informed Neural Network with Mixed Sampling and Its Application in High-Dimensional Systems

Kernel functions embed into the autoencoder to identify the sparse models of nonlinear dynamics

Numerous researches have shown that there are three main challenges in data-driven model identification methods: high-dimensional measurements, system complexity and unknown underlying dynamical properties. For most nonlinear dynamics, the feature space defined by the coefficients of their control equations is sparse. Therefore, sparse regression methods are used to learn the sparse coefficients of the control equations of nonlinear dynamics. However, this method strongly depends on the appropriate selection of the sparse basis vectors. In this essay, the autoencoder is combined with the sparse regression method to simultaneously identify the sparse coordinate and a parsimonious, interpretable and generalizable model of the specified system. It also integrates kernel functions to map the intractable measurements in the hidden space of the autoencoder into a linearly distinguishable kernel space, which kernelizes the candidate function library of the sparse identification of nonlinear dynamics (SINDy) model as the sparse dictionaries. Therefore, the flexible representation of neural networks, the simplicity of sparse regression methods and the implicit non-linear representation of kernel functions are consolidated in this article. To inspect the reliability of the proposed model in this paper, a set of nonlinear dynamics formulated by ordinary differential equations (ODEs), second-order trigonometric functions and partial differential equations (PDEs) are utilized as test cases. And the comparisons between the proposed model and other model identification methods illustrate that the performance of the former is the best.

TGN: A Temporal Graph Network for Physics Prediction

Long-term prediction of physical systems on irregular unstructured meshes is extremely challenging due to the spatial complexityof meshes and the dynamic changes over time; namely, spatial dependence and temporal dependence. Recently, graph-based next-step prediction models have achieved great success in the task of modeling complex high-dimensional physical systems. However, due to these models ignoring the temporal dependence, they inevitably suffer from the effects of error accumulation. To capture the spatial and temporal dependence simultaneously, we propose a temporal graph network (TGN) to predict the long-term dynamics of complex physical systems. Specifically, we introduce an Encode-Process-Decode architecture to capture spatial dependence and create low-dimensional vector representations of system states. Additionally, a temporal model is introduced to learn the dynamic changes in the low-dimensional vector representations to capture temporal dependence. Our model can capture spatiotemporal correlations within physical systems. On some complex long-term prediction tasks in fluid dynamics, such as airfoil flow and cylinder flow, the prediction error of our method is significantly lower than the competitive GNN baseline. We show accurate phase predictions even for very long prediction sequences.