Multiscale Dynamical Systems Research Articles

Detection and Reconstruction of Bursting Oscillations in Complex Systems Using the HAVOK Analysis Framework

Multi-scale dynamical systems may exhibit bursting oscillations, which are typically identified by analyzing time series and phase portraits. However, in cases where bursting oscillations are not apparent, relying solely on these methods may have limitations in accurately detecting their occurrence. This paper introduces the HAVOK analysis framework to the field of bursting oscillations. By using single-variable time series data, models that may produce bursting oscillations are restructured into forced linear models. This approach allows for the rapid prediction of bursting oscillations by observing the forced terms. The results show that the intermittent periodic bursts in the visualizations of the forced eigen time series within the HAVOK framework are strongly correlated with the excitation states in bursting oscillations, enabling the prediction of their occurrence. Especially in cases where it is challenging to determine the presence of bursting oscillations through time series plots alone, this method can still sensitively detect them. Additionally, the embedded and reconstructed flow fields plotted using this approach can help understand the dynamics of bursting oscillations in certain scenarios.

Iterative Ensemble Smoothers for Data Assimilation in Coupled Nonlinear Multiscale Models

Abstract This paper identifies and explains particular differences and properties of adjoint-free iterative ensemble methods initially developed for parameter estimation in petroleum models. The aim is to demonstrate the methods’ potential for sequential data assimilation in coupled and multiscale unstable dynamical systems. For this study, we have introduced a new nonlinear and coupled multiscale model based on two Kuramoto–Sivashinsky equations operating on different scales where a coupling term relaxes the two model variables toward each other. This model provides a convenient testbed for studying data assimilation in highly nonlinear and coupled multiscale systems. We show that the model coupling leads to cross covariance between the two models’ variables, allowing for a combined update of both models. The measurements of one model’s variable will also influence the other and contribute to a more consistent estimate. Second, the new model allows us to examine the properties of iterative ensemble smoothers and assimilation updates over finite-length assimilation windows. We discuss the impact of varying the assimilation windows’ length relative to the model’s predictability time scale. Furthermore, we show that iterative ensemble smoothers significantly improve the solution’s accuracy compared to the standard ensemble Kalman filter update. Results and discussion provide an enhanced understanding of the ensemble methods’ potential implementation and use in operational weather- and climate-prediction systems.

Micro-macro consistency in multiscale modeling: Score-based model assisted sampling of fast/slow dynamical systems.

A valuable step in the modeling of multiscale dynamical systems in fields such as computational chemistry, biology, and materials science is the representative sampling of the phase space over long time scales of interest; this task is not, however, without challenges. For example, the long term behavior of a system with many degrees of freedom often cannot be efficiently computationally explored by direct dynamical simulation; such systems can often become trapped in local free energy minima. In the study of physics-based multi-time-scale dynamical systems, techniques have been developed for enhancing sampling in order to accelerate exploration beyond free energy barriers. On the other hand, in the field of machine learning (ML), a generic goal of generative models is to sample from a target density, after training on empirical samples from this density. Score-based generative models (SGMs) have demonstrated state-of-the-art capabilities in generating plausible data from target training distributions. Conditional implementations of such generative models have been shown to exhibit significant parallels with long-established-and physics-based-solutions to enhanced sampling. These physics-based methods can then be enhanced through coupling with the ML generative models, complementing the strengths and mitigating the weaknesses of each technique. In this work, we show that SGMs can be used in such a coupling framework to improve sampling in multiscale dynamical systems.

Weak averaging principle for multiscale stochastic dynamical systems driven by stable processes

We study the averaging principle for a family of multiscale stochastic dynamical systems. The fast and slow components of the systems are driven by two independent stable Lévy noises, whose stable indexes may be different. The homogenizing index r0 of slow components has a relation with the stable index α1 of the noise of fast components given by 0<r0<2−2/α1. By first studying a nonlocal Poisson equation and then constructing suitable correctors, we obtain that the slow components weakly converge to a Lévy process as the scale parameter goes to zero.

Uniformly accurate nested Picard iterative schemes for nonlinear Schrödinger equation with highly oscillatory potential

The nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP) often appears in many multiscale dynamical systems, where the temporal oscillation causes the major numerical difficulties. Recently, the splitting schemes (Su et al., 2020) were analyzed rigorously for solving the NLSE-OP and the error bounds show that they are only uniformly first-order accurate. This obviously can not meet the requirement of high precision in actual calculation. In this paper, in order to obtain a higher uniform convergence order, we study the nested Picard iterative (NPI) schemes for the NLSE-OP with polynomial nonlinearity. Firstly we propose a uniformly accurate first-order exponential wave integrator (EWI) scheme for arbitrary nonlinearity by integrating the potential exactly. Then we construct a uniformly accurate second-order NPI scheme for the NLSE-OP with polynomial nonlinearity. The schemes are fully explicit and very efficient due to the fast Fourier transform (FFT). We give rigorously error analysis and establish error bounds for the numerical solutions without any CFL-type condition constraint. Numerical experiments prove the correctness of our theoretical analysis and the effectiveness of our schemes. Theoretically, our schemes can be extended to uniformly accurate arbitrary high order for the NLSE-OP.

Multiscale contextual emergence of neural dynamics, cognition and action

There is growing evidence that brain processes involve multiscale overlapping networks and that the mapping between such neural processes and cognitive functions is many-to-many. So, the answer to the question what spatiotemporal scales in the brain are most relevant for cognition, action, experience, etc., is that several inextricably interconnected and integrated scales are relevant. There is also growing evidence that brains and embodied agents (people) are part of “larger” distributed “bio-psycho-social networks.” One cannot fully appreciate what brains do and how they work in isolation from these larger multiscale, multi-level, and multi-faceted “4E” networks (embodied, embedded, extended, and enactive). Nor can one explain human experience, cognition, or action without such an understanding. Establishing these claims is the purpose of this paper. Section 2 will unpack the claim that the brain itself is best viewed as several multiscale, dynamical, multifunctional, coordinated, and fully integrated overlapping networks. Furthermore, such individual brain networks and conscious cognitive agents are embedded in “larger” “4E” dynamical networks. Section 3 argues that the best characterization of such 4E networks is not in terms of mechanistic reduction or modularity, but contextual emergence. Section 4 will draw key connections between contextual emergence and the related work of other philosophers and neuroscientists. Lastly, Section 4 will conclude that conscious cognitive agents are reasonably conceived of as highly non-decomposable “4E” contextually emergent multiscale dynamical systems. In short, people are not brains and brains are not computers.

Small noise asymptotics of multi-scale McKean-Vlasov stochastic dynamical systems

The main aim of this work is to investigate small noise limiting behavior of multi-scale McKean-Vlasov stochastic dynamical systems, where we allow the coefficients depend on the distributions of both slow and fast components. Firstly, the strong convergence in the functional law of large numbers is established by the time discretization scheme. Secondly, in order to characterize the probability of deviations away from the averaged limit, we prove the large deviation principle by the weak convergence approach for McKean-Vlasov equations.

Higher-order approximations in the averaging principle of multiscale systems

We study the asymptotic behavior for multiscale dynamical systems. Using the Poisson equation with parameters, we derive a hierarchy of approximation equations which are able to approximate the slow motion in the Lp-sense and achieve order ɛk/2 with any p⩾1 and k∈N+. This generalizes the averaged equation prescribed by the averaging principle, which results in order ɛ1/2 approximation.

Model Predictive Control of water resources systems: A review and research agenda

Model Predictive Control (MPC) has recently gained increasing interest in the adaptive management of water resources systems due to its capability of incorporating disturbance forecasts into real-time optimal control problems. Yet, related literature is scattered with heterogeneous applications, case-specific problem settings, and results that are hardly generalized and transferable across systems. Here, we systematically review 149 peer-reviewed journal articles published over the last 25 years on MPC applied to water reservoirs, open channels, and urban water networks to identify common trends and open challenges in research and practice. The three water systems we consider are inter-connected, multi-purpose and multi-scale dynamical systems affected by multiple hydro-climatic uncertainties and evolving socioeconomic factors. Our review first identifies four main challenges currently limiting most MPC applications in the water domain: (i) lack of systematic benchmarking of MPC with respect to other control methods; (ii) lack of assessment of the impact of uncertainties on the model-based control; (iii) limited analysis of the impact of diverse forecast types, resolutions, and prediction horizons; (iv) under-consideration of the multi-objective nature of most water resources systems. We then argue that future MPC applications in water resources systems should focus on addressing these four challenges as key priorities for future developments.

Statistical Learning of Nonlinear Stochastic Differential Equations from Nonstationary Time Series using Variational Clustering

Parameter estimation for non-stationary stochastic differential equations (SDE) with an arbitrary non-linear drift and non-linear diffusion is accomplished in combination with a non-parametric clustering methodology. Such a model-based clustering approach includes a quadratic programming (QP) problem with equality and inequality constraints. We couple the QP problem to a closed-form likelihood function approach based on suitable Hermite-expansions to approximate the parameter values of the SDE model. The classification problem provides a smooth indicator function, which enables us to recover the underlying temporal parameter modulation of the one-dimensional SDE. As shown by the numerical examples, the clustering approach recovers a hidden functional relationship between the SDE model parameters and an additional auxiliary process. The study builds upon this functional relationship to develop closed-form, non-stationary, data-driven stochastic models for multiscale dynamical systems in real-world applications.

Analysis of multiscale methods for stochastic dynamical systems driven by [formula omitted]-stable processes

In this paper, we firstly analyze the strong convergence of projective integration method for multiscale stochastic dynamical systems driven by α-stable processes, which is used to estimate the effect that the fast components have on slow ones. Then we obtain the pth moment error bounds between the solution of slow component produced by projective integration method and the solution of effective system with p∈1,α. Finally, we corroborate our analytical results through a specific numerical example.

Averaging principle for stochastic differential equations in the random periodic regime

We present the validity of stochastic averaging principle for non-autonomous slow–fast stochastic differential equations (SDEs) whose fast motions admit random periodic solutions. Our investigation is motivated by some problems arising from multi-scale stochastic dynamical systems, where configurations are time dependent due to nonlinearity of the underlying vector fields and the onset of time dependent random invariant sets. Averaging principle with respect to uniform ergodicity of the fast motion is no longer available in this scenario. The ergodicity of time periodic measures of the fast motion on certain minimal Poincaré section is used to identify the averaging limit.

Resampling with neural networks for stochastic parameterization in multiscale systems

In simulations of multiscale dynamical systems, not all relevant processes can be resolved explicitly. Taking the effect of the unresolved processes into account is important, which introduces the need for parameterizations. We present a machine-learning method, used for the conditional resampling of observations or reference data from a fully resolved simulation. It is based on the probabilistic classification of subsets of reference data, conditioned on macroscopic variables. This method is used to formulate a parameterization that is stochastic, taking the uncertainty of the unresolved scales into account. We validate our approach on the Lorenz 96 system, using two different parameter settings which are challenging for parameterization methods.

Effective Filtering for Multiscale Stochastic Dynamical Systems Driven by Lévy Processes*

The work is about multiscale stochastic dynamical systems driven by Lévy processes. We prove that such a system can be approximated by a low-dimensional system on a random invariant manifold, and the original filter can be also approximated by the reduced low-dimensional filter. Finally, we investigate the reduction for $$\varepsilon =0$$ and obtain that these reduced systems does not approximate these multiscale stochastic dynamical systems.

Kernel Analog Forecasting: Multiscale Test Problems

Data-driven prediction is becoming increasingly widespread as the volume of data available grows and as algorithmic development matches this growth. The nature of the predictions made and the manner in which they should be interpreted depend crucially on the extent to which the variables chosen for prediction are Markovian or approximately Markovian. Multiscale systems provide a framework in which this issue can be analyzed. In this work kernel analog forecasting methods are studied from the perspective of data generated by multiscale dynamical systems. The problems chosen exhibit a variety of different Markovian closures, using both averaging and homogenization; furthermore, settings where scale separation is not present and the predicted variables are non-Markovian are also considered. The studies provide guidance for the interpretation of data-driven prediction methods when used in practice.

Discrete-Time Inference for Slow-Fast Systems Driven by Fractional Brownian Motion

We study statistical inference for small-noise-perturbed multiscale dynamical systems where the slow motion is driven by fractional Brownian motion. We develop statistical estimators for both the Hurst index as well as a vector of unknown parameters in the model based on a single time series of observations from the slow process only. We prove that these estimators are both consistent and asymptotically normal as the amplitude of the perturbation and the time-scale separation parameter go to zero. Numerical simulations illustrate the theoretical results.

Predicting critical transitions in multiscale dynamical systems using reservoir computing.

We study the problem of predicting rare critical transition events for a class of slow-fast nonlinear dynamical systems. The state of the system of interest is described by a slow process, whereas a faster process drives its evolution and induces critical transitions. By taking advantage of recent advances in reservoir computing, we present a data-driven method to predict the future evolution of the state. We show that our method is capable of predicting a critical transition event at least several numerical time steps in advance. We demonstrate the success as well as the limitations of our method using numerical experiments on three examples of systems, ranging from low dimensional to high dimensional. We discuss the mathematical and broader implications of our results.

Inhomogeneous functionals and approximations of invariant distributions of ergodic diffusions: Central limit theorem and moderate deviation asymptotics

The paper studies asymptotics of inhomogeneous integral functionals of an ergodic diffusion process under the effect of discretization. Convergence to the corresponding functionals of the invariant distribution is shown for suitably chosen discretization steps, and the fluctuations are analyzed through central limit theorem and moderate deviation principle. The results will be particularly useful for understanding accuracy of an Euler discretization based numerical scheme for approximating functionals of invariant distribution of an ergodic diffusion. This is an infinite-time horizon problem, and the accuracy of numerical schemes in this context are comparatively much less studied than the ones used for generating approximate trajectories of diffusions over finite time intervals. The potential applications of these results also extend to other areas including mathematical physics, parameter inference of ergodic diffusions and analysis of multiscale dynamical systems with averaging.

The Strong Impact of Weak Horizontal Convergence on Continental Shallow Convection

Abstract Idealized large-eddy simulation (LES) is a basic tool for studying three-dimensional turbulence in the planetary boundary layer. LES is capable of providing benchmark solutions for parameterization development efforts. However, real small-scale atmospheric flows develop in heterogeneous and transient environments with locally varying vertical motions inherent to open multiscale interactive dynamical systems. These variations are often too subtle to detect them by state-of-the-art remote and in situ measurements, and are typically excluded from idealized simulations. The present study addresses the impact of weak [i.e., O(10−6) s−1] short-lived low-level large-scale convergence/divergence perturbations on continental shallow convection. The results show a strong response of shallow nonprecipitating convection to the applied weak large-scale dynamical forcing. Evolutions of CAPE, mean liquid water path, and cloud-top heights are significantly affected by the imposed convergence/divergence. In contrast, evolving cloud-base properties, such as the area coverage and mass flux, are only weakly affected. To contrast those impacts with microphysical sensitivity, the baseline simulations are perturbed assuming different observationally based cloud droplet number concentrations and thus different rainfall. For the tested range of microphysical perturbations, the imposed convergence/divergence provides significantly larger impact than changes in the cloud microphysics. Simulation results presented here provide a stringent test for convection parameterizations, especially important for large-scale models progressing toward resolving some nonhydrostatic effects.

On time-splitting methods for nonlinear Schrödinger equation with highly oscillatory potential

In this work, we consider the numerical solution of the nonlinear Schrödinger equation with a highly oscillatory potential (NLSE-OP). The NLSE-OP is a model problem which frequently occurs in recent studies of some multiscale dynamical systems, where the potential introduces wide temporal oscillations to the solution and causes numerical difficulties. We aim to analyze rigorously the error bounds of the splitting schemes for solving the NLSE-OP to a fixed time. Our theoretical results show that the Lie–Trotter splitting scheme is uniformly and optimally accurate at the first order provided that the oscillatory potential is integrated exactly, while the Strang splitting scheme is not. Our results apply to general dispersive or wave equations with an oscillatory potential. The error estimates are confirmed by numerical results.