High-dimensional Dynamical Systems Research Articles

Model-size reduction for reservoir computing by concatenating internal states through time

Reservoir computing (RC) is a machine learning algorithm that can learn complex time series from data very rapidly based on the use of high-dimensional dynamical systems, such as random networks of neurons, called “reservoirs.” To implement RC in edge computing, it is highly important to reduce the amount of computational resources that RC requires. In this study, we propose methods that reduce the size of the reservoir by inputting the past or drifting states of the reservoir to the output layer at the current time step. To elucidate the mechanism of model-size reduction, the proposed methods are analyzed based on information processing capacity proposed by Dambre et al. (Sci Rep 2:514, 2012). In addition, we evaluate the effectiveness of the proposed methods on time-series prediction tasks: the generalized Hénon-map and NARMA. On these tasks, we found that the proposed methods were able to reduce the size of the reservoir up to one tenth without a substantial increase in regression error.

Interactions of multiple rhythms in a biophysical network of neurons

Neural oscillations, including rhythms in the beta1 band (12–20 Hz), are important in various cognitive functions. Often neural networks receive rhythmic input at frequencies different from their natural frequency, but very little is known about how such input affects the network’s behavior. We use a simplified, yet biophysical, model of a beta1 rhythm that occurs in the parietal cortex, in order to study its response to oscillatory inputs. We demonstrate that a cell has the ability to respond at the same time to two periodic stimuli of unrelated frequencies, firing in phase with one, but with a mean firing rate equal to that of the other. We show that this is a very general phenomenon, independent of the model used. We next show numerically that the behavior of a different cell, which is modeled as a high-dimensional dynamical system, can be described in a surprisingly simple way, owing to a reset that occurs in the state space when the cell fires. The interaction of the two cells leads to novel combinations of properties for neural dynamics, such as mode-locking to an input without phase-locking to it.

Threefold way to the dimension reduction of dynamics on networks: An application to synchronization

Several complex systems can be modeled as large networks in which the state of the nodes continuously evolves through interactions among neighboring nodes, forming a high-dimensional nonlinear dynamical system. One of the main challenges of Network Science consists in predicting the impact of network topology and dynamics on the evolution of the states and, especially, on the emergence of collective phenomena, such as synchronization. We address this problem by proposing a Dynamics Approximate Reduction Technique (DART) that maps high-dimensional (complete) dynamics unto low-dimensional (reduced) dynamics while preserving the most salient features, both topological and dynamical, of the original system. DART generalizes recent approaches for dimension reduction by allowing the treatment of complex-valued dynamical variables, heterogeneities in the intrinsic properties of the nodes as well as modular networks with strongly interacting communities. Most importantly, we identify three major reduction procedures whose relative accuracy depends on whether the evolution of the states is mainly determined by the intrinsic dynamics, the degree sequence, or the adjacency matrix. We use phase synchronization of oscillator networks as a benchmark for our threefold method. We successfully predict the synchronization curves for three phase dynamics (Winfree, Kuramoto, theta) on the stochastic block model. Moreover, we obtain the bifurcations of the Kuramoto-Sakaguchi model on the mean stochastic block model with asymmetric blocks and we show numerically the existence of periphery chimera state on the two-star graph. This allows us to highlight the critical role played by the asymmetry of community sizes on the existence of chimera states. Finally, we systematically recover well-known analytical results on explosive synchronization by using DART for the Kuramoto-Sakaguchi model on the star graph.

Addressing the curse of dimensionality in stochastic dynamics: a Wiener path integral variational formulation with free boundaries.

A Wiener path integral variational formulation with free boundaries is developed for determining the stochastic response of high-dimensional nonlinear dynamical systems in a computationally efficient manner. Specifically, a Wiener path integral representation of a marginal or lower-dimensional joint response probability density function is derived. Due to this a priori marginalization, the associated computational cost of the technique becomes independent of the degrees of freedom (d.f.) or stochastic dimensions of the system, and thus, the 'curse of dimensionality' in stochastic dynamics is circumvented. Two indicative numerical examples are considered for highlighting the capabilities of the technique. The first relates to marine engineering and pertains to a structure exposed to nonlinear flow-induced forces and subjected to non-white stochastic excitation. The second relates to nano-engineering and pertains to a 100-d.f. stochastically excited nonlinear dynamical system modelling the behaviour of large arrays of coupled nano-mechanical oscillators. Comparisons with pertinent Monte Carlo simulation data demonstrate the computational efficiency and accuracy of the developed technique.

High-dimensional dynamic systems identification with additional constraints

This note presents a unified analysis of the identification of dynamical systems with low-rank constraints under high-dimensional scaling. This identification problem for dynamic systems is challenging due to the intrinsic dependency of the data. To alleviate this problem, we first formulate this identification problem into a multivariate linear regression problem with a row-sub-Gaussian measurement matrix using the more general input designs and the independently repeated sampling schemes. We then propose a nuclear norm heuristic method that estimates the parameter matrix of a dynamic system from a few input-state data samples. Based on this, we can extend the existing results. In this article, we consider two scenarios. (i) In the noiseless scenario, nuclear-norm minimization is introduced for promoting low-rank. We define the notion of weak restricted isometry property, which is weaker than the ordinary restricted isometry property, and show it holds with high probability for the row-sub-Gaussian measurement matrix. Thereby, the rank-minimization matrix can be exactly recovered from a finite number of data samples. (ii) In the noisy scenario, a regularized framework involving the nuclear norm penalty is established. We give the notion of operator norm curvature condition for the loss function and show it holds for the row-sub-Gaussian measurement matrix with high probability. Consequently, when specifying the suitable choice of the regularization parameter, the operator norm error of the optimal solution of this program has a sharp bound given a finite amount of data samples. This operator norm error bound is always smaller than the ordinary Frobenius norm error bound obtained in the existing work.

On occurrence of bursting oscillations in a dynamical system with a double Hopf bifurcation and slow-varying parametric excitations

Slow-fast analysis has been extensively used in the past to study the occurrence of bursting oscillations. Most bursting oscillations studies are performed on the low dimensional autonomous systems where only codimension-1 bifurcations take place at the transitions of quiescent and spiking states. However, in high dimensional slow-fast dynamical systems, there exist higher co-dimensional bifurcations, which may lead to much more complicated mechanisms for bursting oscillations. To reveal and understand the complicated phenomena, the present paper investigates the normal form of a four-dimensional cubic order system with parametric excitation as a slow parameter. Parametric frequency is chosen far less than the natural frequencies to make sure the coupling of fast and slow scales. In the absence of parametric excitation, the system has a double Hopf bifurcation at the origin. Several equilibrium branches and mechanisms of bursting oscillations are obtained for different amplitudes of the parametric excitation. The trajectory of the system is projected onto two independent sub-planes, while the movement of bursting oscillators is synthesized by the properties in each sub-plane. In the sub-plane, the mechanism of bursting oscillations is derived by employing the overlap of the transformed phase portrait and the equilibrium branches as well as the bifurcations. The codimension-2 double Hopf bifurcation may result in the occurrence of the bursting oscillations in four dimensions. When the excitation amplitude is relatively small, the trajectory only follows the equilibrium branches of trivial and single-mode solutions. With the increase of the excitation amplitude, the trajectory switches between the equilibrium branches of single-mode and mixed-mode. Furthermore, a few interesting phenomena are presented.

Tipping point and noise-induced transients in ecological networks.

A challenging and outstanding problem in interdisciplinary research is to understand the interplay between transients and stochasticity in high-dimensional dynamical systems. Focusing on the tipping-point dynamics in complex mutualistic networks in ecology constructed from empirical data, we investigate the phenomena of noise-induced collapse and noise-induced recovery. Two types of noise are studied: environmental (Gaussian white) noise and state-dependent demographic noise. The dynamical mechanism responsible for both phenomena is a transition from one stable steady state to another driven by stochastic forcing, mediated by an unstable steady state. Exploiting a generic and effective two-dimensional reduced model for real-world mutualistic networks, we find that the average transient lifetime scales algebraically with the noise amplitude, for both environmental and demographic noise. We develop a physical understanding of the scaling laws through an analysis of the mean first passage time from one steady state to another. The phenomena of noise-induced collapse and recovery and the associated scaling laws have implications for managing high-dimensional ecological systems.

Deep neural networks for nonlinear model order reduction of unsteady flows

Unsteady fluid systems are nonlinear high-dimensional dynamical systems that may exhibit multiple complex phenomena in both time and space. Reduced Order Modeling (ROM) of fluid flows has been an active research topic in the recent decade with the primary goal to decompose complex flows into a set of features most important for future state prediction and control, typically using a dimensionality reduction technique. In this work, a novel data-driven technique based on the power of deep neural networks for ROM of the unsteady fluid flows is introduced. An autoencoder network is used for nonlinear dimension reduction and feature extraction as an alternative for singular value decomposition (SVD). Then, the extracted features are used as an input for a long short-term memory (LSTM) network to predict the velocity field at future time instances. The proposed autoencoder-LSTM method is compared with non-intrusive reduced order models based on dynamic mode decomposition (DMD) and proper orthogonal decomposition. Moreover, an autoencoder-DMD algorithm is introduced for ROM, which uses the autoencoder network for dimensionality reduction rather than SVD rank truncation. The results show that the autoencoder-LSTM method is considerably capable of predicting fluid flow evolution, where higher values for the coefficient of determination R2 are obtained using autoencoder-LSTM compared to other models.

Application of adaptive multilevel splitting to high-dimensional dynamical systems

Stochastic nonlinear dynamical systems can undergo rapid transitions relative to the change in their forcing, for example due to the occurrence of multiple equilibrium solutions for a specific interval of parameters. In this paper, we modify one of the methods developed to compute probabilities of such transitions, Trajectory-Adaptive Multilevel Sampling (TAMS), to be able to apply it to high-dimensional systems. The key innovation is a projected time-stepping approach, which leads to a strong reduction in computational costs, in particular memory usage. The performance of this new implementation of TAMS is studied through an example of the collapse of the Atlantic Ocean Circulation.

Backward Reachability Analysis for Nonlinear Dynamical Systems via Pseudospectral Method

In this paper, we propose a new approach to solving the backward reachability problem for nonlinear dynamical systems. Previously, this class of problems has been studied within frameworks of optimal control and zero-sum differential games, where a backward reachable set can be expressed as the zero sublevel set of the value function that can be characterized by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE). In many cases, however, a high computational cost is incurred in numerically solving such HJB PDEs due to the curse of dimensionality. We use the pseudospectral method to convert the associated optimal control problem into nonlinear programs (NLPs). We then show that the zero sublevel set obtained by the optimal cost of the NLP is the corresponding backward reachable set. Note that our approach does not require solving complex HJB PDEs. Therefore, it can reduce computation time and handle high-dimensional dynamical systems, compared with the numerical software package developed by I. Mitchell, which has been used widely in the literature to obtain backward reachable sets by solving HJB equations. We provide several examples to validate the effectiveness of the proposed approach.

Enhancing subdivision technique with an adaptive interpolation sampling method for global attractors of nonlinear dynamical systems

The cell mapping method is a prominent one for global analysis of nonlinear dynamical systems, with which multiple invariant sets can be obtained. However, it is a continuous challenge to enhance the efficiency of the cell mapping method, especially when dealing with high-dimensional nonlinear dynamical systems. In this paper, the subdivision technique commonly used in the cell mapping method is incorporated with an interpolation sampling method, which can further enhance the efficiency over the set-oriented method with subdivision for global attractors of nonlinear dynamical systems. In the present method, a new lattice of interpolating nodes is adopted to fit into the nesting structures of the subdivided cells, portions of which are sequentially removed when resolution goes from low to high. An improved interpolation method is developed to obtain one-step sample mappings, instead of integrations when error bounds are met, in the process of subdivision iterations. Furthermore, a Hash table is introduced in order to fast search and locate the coordinates of cells that cover the invariant sets. Three examples of nonlinear dynamical systems are presented to demonstrate the enhancement in efficiency and effectiveness of the proposed method, indicating that a computational cost is one half down to one sixth of the previous methods.

Path length statistics in reservoir computers.

Because reservoir computers are high dimensional dynamical systems, designing a good reservoir computer is difficult. In many cases, the designer must search a large nonlinear parameter space, and each step of the search requires simulating the full reservoir computer. In this work, I show that a simple statistic based on the mean path length between nodes in the reservoir computer is correlated with better reservoir computer performance. The statistic predicts the diversity of signals produced by the reservoir computer, as measured by the covariance matrix of the reservoir computer. This statistic by itself is not sufficient to predict reservoir computer performance because not only must the reservoir computer produce a diverse set of signals, it must be well matched to the training signals. Nevertheless, this path length statistic allows the designer to eliminate some network configurations from consideration without having to actually simulate the reservoir computer, reducing the complexity of the design process.

A new scalable algorithm for computational optimal control under uncertainty

We address the design of optimal control strategies for high-dimensional stochastic dynamical systems. Such systems may be deterministic nonlinear systems evolving from random initial states, or systems driven by random parameters or random noise. The objective is to provide a validated new computational capability for optimal control which will be achieved more efficiently than current state-of-the-art methods. The new framework utilizes direct single or multi-shooting discretization, and is based on efficient vectorized gradient computation with adaptable memory management. The algorithm is demonstrated to be scalable to high-dimensional nonlinear control systems with random initial condition and unknown parameters. Numerical applications are presented and discussed for stochastic path planning problems involving models of unmanned aerial and ground vehicles, and for distributed control of a nonlinear advection-reaction-diffusion equation.

Incorporating physical constraints in a deep probabilistic machine learning framework for coarse-graining dynamical systems

Data-based discovery of effective, coarse-grained (CG) models of high-dimensional dynamical systems presents a unique challenge in computational physics and particularly in the context of multiscale problems. The present paper offers a data-based, probabilistic perspective that enables the quantification of predictive uncertainties. One of the outstanding problems has been the introduction of physical constraints in the probabilistic machine learning objectives. The primary utility of such constraints stems from the undisputed physical laws such as conservation of mass, energy etc. that they represent. Furthermore and apart from leading to physically realistic predictions, they can significantly reduce the requisite amount of training data which for high-dimensional, multiscale systems are expensive to obtain (Small Data regime). We formulate the coarse-graining process by employing a probabilistic state-space model and account for the aforementioned equality constraints as virtual observables in the associated densities. We demonstrate how deep neural nets in combination with probabilistic inference tools can be employed to identify the coarse-grained variables and their evolution model without ever needing to define a fine-to-coarse (restriction) projection and without needing time-derivatives of state variables.We advocate a sparse Bayesian learning perspective which avoids overfitting and reveals the most salient features in the CG evolution law. The formulation adopted enables the quantification of a crucial, and often neglected, component in the CG process, i.e. the predictive uncertainty due to information loss. Furthermore, it is capable of reconstructing the evolution of the full, fine-scale system and therefore the observables of interest need not be selected a priori. We demonstrate the efficacy of the proposed framework by applying it to systems of interacting particles and a series of images of a nonlinear pendulum. In both cases we identify the underlying coarse dynamics and can generate extrapolative predictions including the forming and propagation of a shock for the particle systems and a stable trajectory in the phase space for the pendulum.

A General Framework to Increase Safety of Learning Algorithms for Dynamical Systems Based on Region of Attraction Estimation

Although the state-of-the-art learning approaches exhibit impressive results for dynamical systems, only a few applications on real physical systems have been presented. One major impediment is that the intermediate policy during the training procedure may result in behaviors that are not only harmful to the system itself but also to the environment. In essence, imposing safety guarantees for learning algorithms is vital for autonomous systems acting in the real world. In this article, we propose a computationally effective and general safe learning framework, specifically for complex dynamical systems. With a proper definition of the safe region, a supervisory control strategy, which switches the actions applied on the system between the learning-based controller and a predefined corrective controller, is given. A simplified system facilitates the estimation of the safe region for the high-dimensional dynamical system. During the learning phase, the belief of the safe region is updated with the actual execution results of the corrective controller, which in turn enables the learning-based controller to have more freedom in choosing its actions. Two examples are given to demonstrate the performance of the proposed framework, one simple inverted pendulum to illustrate the online adaptation method, and one quadcopter control task to show the overall performance.

Response and reliability analysis of a high-dimensional stochastic system

The efficiency of the probability density evolution method (PDEM) is improved in this paper by embedding the Kullback–Leibler (K–L) relative sensitivity in the response analysis of a stochastic dynamic system. The response reliability obtained and the probability density function of the response peaks are used for ranking to get a reduced set of random variables for the PDEM analysis. The need of complicated point selection technique with the high-dimensional uncertain variables is therefore alleviated. The proposed method is illustrated with the response analysis of a random crowd-structure system where the load randomness is considered. The acceleration response induced by the presence of the crowd is evaluated with the proposed method. Results obtained highlight the significant improvements in the computation efficiency of the probabilistic response analysis of a high-dimensional dynamic system.

Semi-Implicit and Semi-Explicit Adams-Bashforth-Moulton Methods

Multistep integration methods are widespread in the simulation of high-dimensional dynamical systems due to their low computational costs. However, the stability of these methods decreases with the increase of the accuracy order, so there is a known room for improvement. One of the possible ways to increase stability is implicit integration, but it consequently leads to sufficient growth in computational costs. Recently, the development of semi-implicit techniques achieved great success in the construction of highly efficient single-step ordinary differential equations (ODE) solvers. Thus, the development of multistep semi-implicit integration methods is of interest. In this paper, we propose the simple solution to increase the numerical efficiency of Adams-Bashforth-Moulton predictor-corrector methods using semi-implicit integration. We present a general description of the proposed methods and explicitly show the superiority of ODE solvers based on semi-implicit predictor-corrector methods over their explicit and implicit counterparts. To validate this, performance plots are given for simulation of the van der Pol oscillator and the Rossler chaotic system with fixed and variable stepsize. The obtained results can be applied in the development of advanced simulation software.

Path Integral Policy Improvement With Population Adaptation.

Path integral policy improvement (PI2) is known to be an efficient reinforcement learning algorithm, particularly, if the target system is a high-dimensional dynamical system. However, PI2, and its existing extensions, have adjustable parameters, on which the efficiency depends significantly. This article proposes an extension of PI2 that adjusts all of the critical parameters automatically. Motion acquisition tasks for three different types of simulated legged robots were performed to test the efficacy of the proposed algorithm. The results show that the proposed method cannot only eliminate the burden on the user to set the parameters appropriately but also improve the optimization performance significantly. For one of the acquired motions, a real robot experiment was conducted to show the validity of the motion.

A highly-efficient method for stationary response of multi-degree-of-freedom nonlinear stochastic systems

Analytical and numerical studies of multi-degree-of-freedom (MDOF) nonlinear stochastic or deterministic dynamic systems have long been a technical challenge. This paper presents a highly-efficient method for determining the stationary probability density functions (PDFs) of MDOF nonlinear systems subjected to both additive and multiplicative Gaussian white noises. The proposed method takes advantages of the sufficient conditions of the reduced Fokker-Planck-Kolmogorov (FPK) equation when constructing the trial solution. The assumed solution consists of the analytically constructed trial solutions satisfying the sufficient conditions and an exponential polynomial of the state variables, and delivers a high accuracy of the solution because the analytically constructed trial solutions capture the main characteristics of the nonlinear system. We also make use of the concept from the data-science and propose a symbolic integration over a hypercube to replace the numerical integrations in higher-dimensional space, which has been regarded as the insurmountable difficulty in the classical method of weighted residuals or stochastic averaging for high-dimensional dynamic systems. Three illustrative examples of MDOF nonlinear systems are analyzed in detail. The accuracy of the numerical results is validated by comparison with the Monte Carlo simulation (MCS) or the available exact solution. Furthermore, we also show the substantial gain in the computational efficiency of the proposed method compared with the MCS.

Extension, embedding and global stability in two dimensional monotone maps

We consider the general second order difference equation \begin{document}$ x_{n+1} = F(x_n, x_{n-1}) $\end{document} in which \begin{document}$ F $\end{document} is continuous and of mixed monotonicity in its arguments. In equations with negative terms, a persistent set can be a proper subset of the positive orthant, which motivates studying global stability with respect to compact invariant domains. In this paper, we assume that \begin{document}$ F $\end{document} has a semi-convex compact invariant domain, then make an extension of \begin{document}$ F $\end{document} on a rectangular domain that contains the invariant domain. The extension preserves the continuity and monotonicity of \begin{document}$ F. $\end{document} Then we use the embedding technique to embed the dynamical system generated by the extended map into a higher dimensional dynamical system, which we use to characterize the asymptotic dynamics of the original system. Some illustrative examples are given at the end.