Nonlinear Spatiotemporal Systems Research Articles

Neural Network Boundary Approximation for Uncertain Nonlinear Spatiotemporal Systems and Its Application of Tracking Control.

This brief addresses the neural network (NN) approximation problem for uncertain nonlinear systems with time-varying parameters (that is, unknown nonlinear spatiotemporal systems). Due to the fact that the unknown spatiotemporal functions cannot be directly approximated by NNs, a so-called time-varying parameter extraction is given to separate time-varying parameters from uncertain nonlinear spatiotemporal functions. By using the supremum of Euler norm of the extracted time-varying parameters, the nonlinear spatiotemporal function is mapped to an unknown state-based boundary function, which can be approximated by NNs. Based on the time-varying parameter extraction, an adaptive neural tracking control law is designed for uncertain strict-feedback nonlinear spatiotemporal systems, which guarantees the convergence of the tracking error with a trajectory performance. The effectiveness of the designed method is verified by simulations.

Nonlinear system dynamics of calcium and nitric oxide due to cell memory and superdiffusion in neurons

The integer-order interdependent calcium ([Ca2+]) and nitric oxide (NO) systems are unable to shed light on the influences of the superdiffusion and memory in triggering Brownian motion (BM) in neurons. Therefore, a mathematical model is constructed for the fractional-order nonlinear spatiotemporal systems of [Ca2+] and NO incorporating reaction-diffusion equations in neurons. The two-way feedback process between [Ca2+] and NO systems through calcium feedback on NO production and NO feedback on calcium through cyclic guanosine monophosphate (cGMP) with plasmalemmal [Ca2+]-ATPase (PMCA) was incorporated in the model. The Crank–Nicholson scheme (CNS) with Grunwald approximation along spatial derivatives and L1 scheme along temporal derivatives with Gauss–Seidel (GS) iterations were employed. The numerical outcomes were analyzed to get insights into superdiffusion, buffer, and memory exhibiting BM of [Ca2+] and NO systems. The conditions, events and mechanisms leading to dysfunctions in calcium and NO systems and causing different diseases like Parkinson’s were explored in neurons.

Decentralized digital twins of complex dynamical systems

In this article, we introduce a decentralized digital twin (DDT) modeling framework and its potential applications in computational science and engineering. The DDT methodology is based on the idea of federated learning, a subfield of machine learning that promotes knowledge exchange without disclosing actual data. Clients can learn an aggregated model cooperatively using this method while maintaining complete client-specific training data. We use a variety of dynamical systems, which are frequently used as prototypes for simulating complex transport processes in spatiotemporal systems, to show the viability of the DDT framework. Our findings suggest that constructing highly accurate decentralized digital twins in complex nonlinear spatiotemporal systems may be made possible by federated machine learning.

Physics informed WNO

Deep neural operators are recognized as an effective tool for learning solution operators of complex partial differential equations (PDEs). As compared to laborious analytical and computational tools, a single neural operator can predict solutions of PDEs for varying initial or boundary conditions and different inputs without repeated independent runs from scratch. A recently proposed Wavelet Neural Operator (WNO) is one such operator that harnesses the advantage of time–frequency localization of wavelets to capture the manifolds in the spatial domain effectively. While WNO has proven to be a promising method for operator learning, the data-hungry nature of the framework is a major shortcoming. Relying completely on conventional solvers for data generation and subsequently training operators with generated data leads to a time-consuming and challenging implementation of the operators in practical applications. In this work, we propose a physics-informed WNO for learning the solution operators of families of parametric PDEs without labeled training data. The efficacy of the framework is validated and illustrated with four nonlinear spatiotemporal systems relevant to various fields of engineering and science.

Variational multiscale reinforcement learning for discovering reduced order closure models of nonlinear spatiotemporal transport systems

A central challenge in the computational modeling and simulation of a multitude of science applications is to achieve robust and accurate closures for their coarse-grained representations due to underlying highly nonlinear multiscale interactions. These closure models are common in many nonlinear spatiotemporal systems to account for losses due to reduced order representations, including many transport phenomena in fluids. Previous data-driven closure modeling efforts have mostly focused on supervised learning approaches using high fidelity simulation data. On the other hand, reinforcement learning (RL) is a powerful yet relatively uncharted method in spatiotemporally extended systems. In this study, we put forth a modular dynamic closure modeling and discovery framework to stabilize the Galerkin projection based reduced order models that may arise in many nonlinear spatiotemporal dynamical systems with quadratic nonlinearity. However, a key element in creating a robust RL agent is to introduce a feasible reward function, which can be constituted of any difference metrics between the RL model and high fidelity simulation data. First, we introduce a multi-modal RL to discover mode-dependant closure policies that utilize the high fidelity data in rewarding our RL agent. We then formulate a variational multiscale RL (VMRL) approach to discover closure models without requiring access to the high fidelity data in designing the reward function. Specifically, our chief innovation is to leverage variational multiscale formalism to quantify the difference between modal interactions in Galerkin systems. Our results in simulating the viscous Burgers equation indicate that the proposed VMRL method leads to robust and accurate closure parameterizations, and it may potentially be used to discover scale-aware closure models for complex dynamical systems.

Two-color optically addressed spatial light modulator as a generic spatiotemporal system.

Nonlinear spatiotemporal systems are the basis for countless physical phenomena in such diverse fields as ecology, optics, electronics, and neuroscience. The canonical approach to unify models originating from different fields is the normal form description, which determines the generic dynamical aspects and different bifurcation scenarios. Realizing different types of dynamical systems via one experimental platform that enables continuous transition between normal forms through tuning accessible system parameters is, therefore, highly relevant. Here, we show that a transmissive, optically addressed spatial light modulator under coherent optical illumination and optical feedback coupling allows tuning between pitchfork, transcritical, and saddle-node bifurcations of steady states. We demonstrate this by analytically deriving the system's dynamical equations in correspondence to the normal forms of the associated bifurcations and confirm these results via extensive numerical simulations. Our model describes a nematic liquid crystal device using nano-dimensional dichalcogenide (a-As 2S 3) glassy thin films as photo sensors and alignment layers, and we use device parameters obtained from experimental characterization. Optical coupling, for example, using diffraction, holography, or integrated unitary maps allows implementing a variety of system topologies of technological relevance for neural networks and potentially Ising or XY-Hamiltonian models with ultralow energy consumption.

Physics-informed learning of governing equations from scarce data

Harnessing data to discover the underlying governing laws or equations that describe the behavior of complex physical systems can significantly advance our modeling, simulation and understanding of such systems in various science and engineering disciplines. This work introduces a novel approach called physics-informed neural network with sparse regression to discover governing partial differential equations from scarce and noisy data for nonlinear spatiotemporal systems. In particular, this discovery approach seamlessly integrates the strengths of deep neural networks for rich representation learning, physics embedding, automatic differentiation and sparse regression to approximate the solution of system variables, compute essential derivatives, as well as identify the key derivative terms and parameters that form the structure and explicit expression of the equations. The efficacy and robustness of this method are demonstrated, both numerically and experimentally, on discovering a variety of partial differential equation systems with different levels of data scarcity and noise accounting for different initial/boundary conditions. The resulting computational framework shows the potential for closed-form model discovery in practical applications where large and accurate datasets are intractable to capture.

Inference on high-dimensional implicit dynamic models using a guided intermediate resampling filter.

We propose a method for inference on moderately high-dimensional, nonlinear, non-Gaussian, partially observed Markov process models for which the transition density is not analytically tractable. Markov processes with intractable transition densities arise in models defined implicitly by simulation algorithms. Widely used particle filter methods are applicable to nonlinear, non-Gaussian models but suffer from the curse of dimensionality. Improved scalability is provided by ensemble Kalman filter methods, but these are inappropriate for highly nonlinear and non-Gaussian models. We propose a particle filter method having improved practical and theoretical scalability with respect to the model dimension. This method is applicable to implicitly defined models having analytically intractable transition densities. Our method is developed based on the assumption that the latent process is defined in continuous time and that a simulator of this latent process is available. In this method, particles are propagated at intermediate time intervals between observations and are resampled based on a forecast likelihood of future observations. We combine this particle filter with parameter estimation methodology to enable likelihood-based inference for highly nonlinear spatiotemporal systems. We demonstrate our methodology on a stochastic Lorenz 96 model and a model for the population dynamics of infectious diseases in a network of linked regions.

Modified Empirical Eigenfunctions and Its Applications for Model Reduction of Nonlinear Spatiotemporal Systems

Model reduction can greatly reduce complexity and difficulty of control design for spatiotemporal systems (STS) in engineering applications. Empirical eigenfunctions (EEFs) are widely used for the model reduction of spatiotemporal systems, however, truncation of higher modes may describe the behaviours of nonlinear spatiotemporal systems inaccurately. In this paper, modified EEFs are proposed and applied to model reduction of nonlinear spatiotemporal systems. Modified EEFs are obtained via modifying the weights matrix in the method of snapshots, which can be rewritten as linear combinations of initial EEFs. The coefficient matrix for combinations is computed according to the nonlinear temporal dynamics of STSs. Thus, the effects of higher modes are considered into modified EEFs with less computational requirements. The reduced model can give a more accurate description for behaviours of the system. The performance of the proposed method is further proved theoretically, and a numerical example demonstrates the effectiveness of the proposed method.

Local causal states and discrete coherent structures.

Coherent structures form spontaneously in nonlinear spatiotemporal systems and are found at all spatial scales in natural phenomena from laboratory hydrodynamic flows and chemical reactions to ocean, atmosphere, and planetary climate dynamics. Phenomenologically, they appear as key components that organize the macroscopic behaviors in such systems. Despite a century of effort, they have eluded rigorous analysis and empirical prediction, with progress being made only recently. As a step in this, we present a formal theory of coherent structures in fully discrete dynamical field theories. It builds on the notion of structure introduced by computational mechanics, generalizing it to a local spatiotemporal setting. The analysis' main tool employs the local causal states, which are used to uncover a system's hidden spatiotemporal symmetries and which identify coherent structures as spatially localized deviations from those symmetries. The approach is behavior-driven in the sense that it does not rely on directly analyzing spatiotemporal equations of motion, rather it considers only the spatiotemporal fields a system generates. As such, it offers an unsupervised approach to discover and describe coherent structures. We illustrate the approach by analyzing coherent structures generated by elementary cellular automata, comparing the results with an earlier, dynamic-invariant-set approach that decomposes fields into domains, particles, and particle interactions.

A Spatial Frequency Domain Analysis of the Belousov–Zhabotinsky Reaction

A nonlinear spatio-temporal system identification and analysis approach is used to study the dynamical behavior of the Belousov–Zhabotinsky (BZ) chemical reaction process. In our previous study [Guo et al., 2010],, the dynamical behavior of the BZ reaction in the spatial-temporal domain has been analyzed by identifying a coupled map lattice (CML) model of the process directly from experimental data from a real BZ reaction experiment. In this paper, the frequency domain analysis of the dynamics near equilibrium is carried out by mapping the obtained CML model into higher order spatial frequency response functions to reveal the nonlinear coupling and modulation between the various initial spectral components in the process. As far as we are aware, this is the first study of any real spatio-temporal system using a spatio-temporal domain identification and frequency domain analysis approach.

Online Identification of Nonlinear Spatiotemporal Systems Using Kernel Learning Approach

The identification of nonlinear spatiotemporal systems is of significance to engineering practice, since it can always provide useful insight into the underlying nonlinear mechanism and physical characteristics under study. In this paper, nonlinear spatiotemporal system models are transformed into a class of multi-input-multi-output (MIMO) partially linear systems (PLSs), and an effective online identification algorithm is therefore proposed by using a pruning error minimization principle and least square support vector machines. It is shown that many benchmark physical and engineering systems can be transformed into MIMO-PLSs which keep some important physical spatiotemporal relationships and are very helpful in the identification and analysis of the underlying system. Compared with several existing methods, the advantages of the proposed method are that it can make full use of some prior structural information about system physical models, can realize online estimation of the system dynamics, and achieve accurate characterization of some important nonlinear physical characteristics of the system. This would provide an important basis for state estimation, control, optimal analysis, and design of nonlinear distributed parameter systems. The proposed algorithm can also be applied to identification problems of stochastic spatiotemporal dynamical systems. Numeral examples and comparisons are given to demonstrate our results.

Spatiotemporal system identification on nonperiodic domains using Chebyshev spectral operators and system reduction algorithms

A system identification methodology based on Chebyshev spectral operators and an orthogonal system reduction algorithm is proposed, leading to a new approach for data-driven modeling of nonlinear spatiotemporal systems on nonperiodic domains. A continuous model structure is devised allowing for terms of arbitrary derivative order and nonlinearity degree. Chebyshev spectral operators are introduced to realm of inverse problems to discretize that continuous structure and arrive with spectral accuracy at a discrete form. Finally, least squares combined with an orthogonal system reduction algorithm are employed to solve for the parameters and eliminate the redundancies to achieve a parsimonious model. A numerical case study of identifying the Allen-Cahn metastable equation demonstrates the superior accuracy of the proposed Chebyshev spectral identification over its finite difference counterpart.

A signal estimation approach to functional MRI.

In the last half decade, fast methods of magnetic resonance imaging have led to the possibility, for the first time, of non-invasive dynamic brain imaging. This has led to an explosion of work in the Neurosciences. From a signal processing viewpoint the problems are those of nonlinear spatio-temporal system identification. In this paper, we develop new methods of identification using novel spatial regularization. We also develop a new model comparison technique and use that to compare our method with existing techniques on some experimental data.

Growth dynamics and morphology of passive films

We study the passive layer formation observed when a metal is immersed in an oxidizing solvent and propose a model based on two different schemes to describe the evolution of the solid film and the liquid phase. The formation of the passive layer is simulated at a mesoscopic scale, whereas the concentrations in soluble species in the liquid are deduced from the numerical resolution of a macroscopic equation. The evolution of the two phases is coupled by the particular boundary conditions imposed at the moving interface. The simulation model involves reaction and diffusion processes, including a poisoning of the traveling interface due to chemical species initially added to the solvent or produced by the growth reaction itself. Depending on the parameter values, in particular on the production of poison during the growth reaction, different asymptotic behaviors are reached. The simulation brings out the existence of a large parameter domain in which the system spontaneously evolves toward a critical state. This behavior follows a short transient regime associated with the growth of a thin compact structure. The existence of such a primary layer in contact with the metal has been invoked in the literature to explain experimental results on lithium batteries. In stationary conditions, the layer observed is porous and characterized by a self-similar geometry. We find an order parameter whose fluctuations are controlled by the diffusion coefficient of the poisoning species and have a power-law behavior. The concept of self-organized criticality has been proposed to unify such a dynamical regulation around a critical state in an open nonlinear spatiotemporal system. We believe that the growth of thick films offers an example of this behavior. \textcopyright{} 1996 The American Physical Society.