Equation-free Method Research Articles

Quantifying changes in individual-specific template-based representations of center-of-mass dynamics during walking with ankle exoskeletons using Hybrid-SINDy

Ankle exoskeletons alter whole-body walking mechanics, energetics, and stability by altering center-of-mass (CoM) motion. Controlling the dynamics governing CoM motion is, therefore, critical for maintaining efficient and stable gait. However, how CoM dynamics change with ankle exoskeletons is unknown, and how to optimally model individual-specific CoM dynamics, especially in individuals with neurological injuries, remains a challenge. Here, we evaluated individual-specific changes in CoM dynamics in unimpaired adults and one individual with post-stroke hemiparesis while walking in shoes-only and with zero-stiffness and high-stiffness passive ankle exoskeletons. To identify optimal sets of physically interpretable mechanisms describing CoM dynamics, termed template signatures, we leveraged hybrid sparse identification of nonlinear dynamics (Hybrid-SINDy), an equation-free data-driven method for inferring sparse hybrid dynamics from a library of candidate functional forms. In unimpaired adults, Hybrid-SINDy automatically identified spring-loaded inverted pendulum-like template signatures, which did not change with exoskeletons (p > 0.16), except for small changes in leg resting length (p < 0.001). Conversely, post-stroke paretic-leg rotary stiffness mechanisms increased by 37–50% with zero-stiffness exoskeletons. While unimpaired CoM dynamics appear robust to passive ankle exoskeletons, how neurological injuries alter exoskeleton impacts on CoM dynamics merits further investigation. Our findings support Hybrid-SINDy’s potential to discover mechanisms describing individual-specific CoM dynamics with assistive devices.

Detection of the onset of yielding and creep failure from digital image correlation

There are a multitude of applications in which structural materials would be desired to be nondestructively evaluated, while in a component, for plasticity and failure characteristics. In this way, safety and resilience features can be significantly improved. Nevertheless, while failure can be visible through cracks, plasticity is commonly invisible and highly microstructure-dependent. Here, we show that an equation-free method based on principal component analysis is capable of detecting yielding and tertiary creep onset, directly from strain fields that are obtained by digital image correlation, applicable on components, continuously and nondestructively. We demonstrate the applicability of the method to yielding of Ni-based Haynes 230 metal alloy polycrystalline samples, which are also characterized through electron microscopy and benchmarked using continuum polycrystalline plasticity modeling. Also, we successfully apply this method to yielding during uniaxial tension of Hastelloy X polycrystalline samples, and also to the onset of tertiary creep in quasibrittle fiber composites under uniaxial tension. We conclude that there are key features in the spatiotemporal fluctuations of local strain fields that can be used to infer mechanical properties.

Optimal vaccine roll-out strategies including social distancing for pandemics

SummaryNon-pharmacological interventions (NPIs), principally social distancing, in combination with effective vaccines, aspire to develop a protective immunity shield against pandemics and particularly against the COVID-19 pandemic. In this study, an agent-based network model with small-world topology is employed to find optimal policies against pandemics, including social distancing and vaccination strategies. The agents’ states are characterized by a variation of the SEIR model (susceptible, exposed, infected, recovered). To explore optimal policies, an equation-free method is proposed to solve the inverse problem of calibrating an agent’s infection rate with respect to the vaccination efficacy. The results show that prioritizing the first vaccine dose in combination with mild social restrictions, is sufficient to control the pandemic, with respect to the number of deaths. Moreover, for the same mild number of social contacts, we find an optimal vaccination ratio of 0.85 between older people of ages compared to younger ones.

Tomographic Particle Image Velocimetry and Dynamic Mode Decomposition (DMD) in a Rectangular Impinging Jet: Vortex Dynamics and Acoustic Generation

Impinging jets are encountered in ventilation systems and many other industrial applications. Their flows are three-dimensional, time-dependent, and turbulent. These jets can generate a high level of noise and often present a source of discomfort in closed areas. In order to reduce and control such mechanisms, one should investigate the flow dynamics that generate the acoustic field. The purpose of this study is to investigate the flow dynamics and, more specifically, the coherent structures involved in the acoustic generation of these jets. Model reduction techniques are commonly used to study the underlying mechanisms by decomposing the flow into coherent structures. The dynamic mode decomposition (DMD) is an equation-free method that relies only on the system’s data taken either through experiments or through numerical simulations. In this paper, the DMD technique is applied, and the spatial modes and their frequencies are presented. The temporal content of the DMD’s modes is then correlated with the acoustic signal. The flow is generated by a rectangular jet impinging on a slotted plate (for a Reynolds number Re = 4458) and its kinematic field is obtained via the tomographic particle image velocimetry technique (TPIV). The findings of this research highlight the coherent structures signature in the DMD’s spectral content and show the cross correlations between the DMD’s modes and the acoustic field.

One-parametric bifurcation analysis of data-driven car-following models

In this study, an equation-free method is used to perform bifurcation analyses of various artificial neural network (ANN) based car-following models. The ANN models were trained on Multiple Car Following (MCF) model output data (ANN-m) and field data (ANN-r). The ANN-m model could capture the behaviour of the MCF model in quite detail. A bifurcation analysis, using the circuit length L as parameter, for the ANN-m model leads to good results if the training data set from the MCF model is sufficiently diverse, namely that it incorporates data from a wide range of vehicle densities that encompass the stable free-flow and the stable jam-flow regimes. The ANN-r model is in general able to capture the feature of traffic jams when a car takes headway and velocity of itself and of the two cars ahead as input. However, the traffic flow of the ANN-r model is more regular in comparison to the field data. It is possible to construct a partial bifurcation diagram in L for the ANN-r using the equation-free method and it is found that the flow changes stability due to a subcritical Hopf bifurcation.

Novel data-driven, equation-free method captures spatio-temporal patterns of neurodegeneration in Parkinson's disease: Application of dynamic mode decomposition to PET.

Most neurodegenerative disorders are characterized by progressive loss of neurons throughout the course of disease in the form of specific spatio-temporal patterns. To capture and quantify these coherent patterns across both space and time, traditionally one would either fit a pre-defined model with spatial and temporal parameters or apply analysis in the spatial and temporal domains separately. In this work, we introduce and validate the use of dynamic mode decomposition (DMD), a data-driven multivariate approach, to extract coupled spatio-temporal patterns simultaneously. We apply the method to examine progressive dopaminergic degeneration in 41 patients with Parkinson's disease (PD) using [11C](±)dihydrotetrabenazine (DTBZ) Positron Emission Tomography (PET). DMD decomposed the progressive dopaminergic changes in the putamen into two orthogonal temporal progression curves associated with distinct spatial patterns: 1) an anterior-posterior gradient, the expression of which decreased gradually with disease progression with a higher initial expression in the less affected side; 2) a dorsal-ventral gradient in the less affected side, which was present in early disease stage only. In the caudate, we found a head-tail gradient analogous to the anterior-posterior gradient seen in the putamen; as in the putamen, the expression of this gradient decreased gradually with disease progression with higher expression in the less affected side. Our results with DTBZ PET data show the applicability and relevance of the proposed method for extracting spatio-temporal patterns of neurotransmitter changes due to neurodegeneration. The method is able to decompose known PD-induced dopaminergic denervation into orthogonal (and thus loosely independent) temporal curves, which may be able to reflect and separate either different mechanisms underlying disease progression and disease initiation, or differential involvement of striatal sub-regions at different disease stages, in a completely data driven way. It is expected that this method can be easily extended to other PET tracers and neurodegenerative disorders and may help to elucidate disease mechanisms in more details compared to traditional approaches.

Dynamic mode decomposition with exogenous input for data-driven modeling of unsteady flows

This work proposes a data-driven reduced-order modeling algorithm for complex, high-dimensional, and unsteady fluid systems with exogenous input and control. This algorithm is a variant of dynamic mode decomposition (DMD), which is an equation-free method for identifying coherent structures and modeling complex flow dynamics. Compared with existing methods, the proposed method improves the capability of predicting the flow evolution near the unstable equilibrium state. The method is achieved by two steps. First, the system matrix without input is identified by standard DMD to represent the intrinsic flow dynamics. Second, the input term, represented by a state space equation, is identified through existing methods for DMD with control effects. The whole system with input is described by the superposition of both the system matrix and the input term. The proposed method is validated by one simple two-dimensional dynamic system and two test cases of unsteady flow, including flow past a circular cylinder at Reynolds number 45 and flow past a NACA0012 airfoil at an angle of attack 25°. Results indicate that the proposed method gives more accurate description on the flow evolution with or without external forcing.

Spectral Methods for Multiscale Stochastic Differential Equations

This paper presents a new method for the solution of multiscale stochastic differential equations at the diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paper we introduce a new numerical methodology that is based on a spectral method. In particular, we use an expansion in Hermite functions to approximate the solution of an appropriate Poisson equation, which is used in order to calculate the coefficients of the homogenized equation. Spectral convergence is proved under suitable assumptions. Numerical experiments corroborate the theory and illustrate the performance of the method. A comparison with the HMM and an application to singularly perturbed stochastic PDEs are also presented.

Nonlinear Model Order Reduction via Dynamic Mode Decomposition

We propose a new technique for obtaining reduced order models for nonlinear dynamical systems. Specifically, we advocate the use of the recently developed dynamic mode decomposition (DMD), an equation-free method, to approximate the nonlinear term. DMD is a spatio-temporal matrix decomposition of a data matrix that correlates spatial features while simultaneously associating the activity with periodic temporal behavior. With this decomposition, one can obtain a fully reduced dimensional surrogate model and avoid the evaluation of the nonlinear term in the online stage. This allows for a reduction in the computational cost and, at the same time, accurate approximations of the problem. We present a suite of numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches.

Nuclear norm regularised dynamic mode decomposition

As a data-driven, equation-free decomposition method, the DMD can characterise dynamic behaviour of a non-linear system by using the DMD modes and eigenvalues. However, all current provable algorithms suffer from a separate procedure for obtaining the DMD modes and determining the number of modes. In this study, the authors propose a nuclear norm regularised DMD (NNR-DMD) algorithm that produces low-dimensional spatio-temporal modes. A nuclear norm regularisation term is added to the optimisation problem of the standard DMD algorithm for prompting the sparsity of the projected DMD modes. Split Bregman method is applied to solve the regularised convex, but non-smooth optimisation problem. Several numerical examples demonstrate the potential of the proposed NNR-DMD algorithm: (i) it can identify the low-dimensional spatio-temporal DMD modes in which each of them possesses a single temporal frequency; (ii) the reconstruction errors based on the sparse DMD modes can be reduced when it compares with the sparsity-promoting DMD algorithm penalising the l 1 -norm of the vector of DMD amplitudes; and (iii) it can obtain low-dimensional coherent structures when the NNR-DMD algorithm is applied to coherency identification of generators in an interconnected power system.

Telescopic projective Adams multiscale modeling of electrochemical reactions in tubular solid oxide fuel cells

Abstract In order to predict the electrochemical performance of Solid Oxide Fuel Cells (SOFCs), a telescopic projective Adams (TPA) multiscale simulation method is proposed in this work. This method is constructed on the basis of the equation-free method (EFM). A lattice Boltzmann model is used as the fine-scale simulator of the proposed method. The electrochemical reaction-diffusion process was simulated by the TPA and the lattice Boltzmann method (LBM). The results of the two methods were found to be in good agreement, and the TPA method can give accurate results with lower computational costs. The electrochemical reactions were also simulated based on the TPA method. The results were consistent with the experimental data, indicating that the proposed TPA method is an effective tool to simulate the electrochemical reactions of SOFCs. Also, the proposed method is suggested to be helpful in multiscale modeling of other energy systems.

Simulation of liquid–vapour phase transitions and multiphase flows by an improved lattice Boltzmann model

An improved lattice Boltzmann (LB) model with a new scheme for the interparticle interaction force term is proposed in this paper. Based on the improved LB model, the equation-free method is implemented for simulating liquid–vapour phase change and multiphase flows. The details of phase separation are presented by numerical simulation results in terms of coexistence curves and spurious currents. Compared with existing models, the proposed model can give more accurate results in a wider temperature range with the spurious currents reduced and less time consumed. Characteristics of phase separation can be quickly and accurately reflected by the proposed method. Then, the contact angle of the solid surface is numerically investigated based on the proposed model. The proposed model is valid for steady flow with near zero velocity; unsteady cases will be investigated in further studies. This work will be helpful for our long-term aim of multi-scale modelling of convective boiling.

方程式フリー法を用いた変形体の動弾性解析

方程式フリー法を用いた変形体の動弾性解析

Implicit Methods for Equation-Free Analysis: Convergence Results and Analysis of Emergent Waves in Microscopic Traffic Models

We introduce a general formulation for an implicit equation-free method in the setting of slow-fast systems. First, we give a rigorous convergence result for equation-free analysis showing that the implicitly defined coarse-level time stepper converges to the true dynamics on the slow manifold within an error that is exponentially small with respect to the small parameter measuring time scale separation. Second, we apply this result to the idealized traffic modeling problem of phantom jams generated by cars with uniform behavior on a circular road. The traffic jams are waves that travel slowly against the direction of traffic. Equation-free analysis enables us to investigate the behavior of the microscopic traffic model on a macroscopic level. The standard deviation of cars' headways is chosen as the macroscopic measure of the underlying dynamics such that traveling wave solutions correspond to equilibria on the macroscopic level in the equation-free setup. The collapse of the traffic jam to the free flow then corresponds to a saddle-node bifurcation of this macroscopic equilibrium. We continue this bifurcation in two parameters using equation-free analysis.

Dynamics of chemical reacting systems in low dimensional lattices via the Equation Free Method

In this report we show how it is possible to reconstruct the dynamics of a chemical reacting systems on lattice via an appropriate choice of macroscopic variable set. When chemical reactions fire on a low dimensional setting the usual mean field approximation (MF) may fail to accurately describe the proper dynamics of the system. Consequently it is not possible to use the concentrations of different species as sole quantities to describe macroscopically such systems and more involved approximations are needed. Traditional approaches introduces higher order correlation functions as extra macroscopic variables, however such expansions usually require closure procedures and tend to quickly demand high number of terms even for simple systems. It is also possible to describe the system in a microscopic way via a Kinetic Monte Carlo (KMC) algorithm, but this approach requires much longer computation time. In this report we study such a problem from a different perspective using the Equation Free Method (EFM) framework; with such approach it is possible to “bridge” between the macroscopic dynamics (mean field) and the microscopic one (KMC). We analyze different techniques to move between these two descriptions of the system trying to understand what information needs to be preserved at the macroscopic level in order to recover correctly all the features of the microscopic dynamics.

Towards an efficient multiscale modeling of low-dimensional reactive systems: Study of numerical closure procedures

We study a numerical closure approach for systems of chemically reacting systems [corrected] on lattices with low-dimensional support, for which a mean-field approximation is insufficiently accurate because of lateral interaction on the lattice. We introduce a hierarchy of macroscopic state variables, taking particle clusters into account, whose time evolution is obtained via microscopic (kinetic Monte Carlo) simulation. The macroscopic state variables are chosen such that they can be [corrected] straightforwardly conserved during reconstruction of a microscopic configuration (the so-called lifting step). We present and compare the effects of different alternatives to initialize the remaining degrees of freedom. We illustrate the strong interplay between the number of macroscopic state variables and the specifics of the lifting and that, for a given lifting operator, accuracy of the macroscopic dynamics does not necessarily improve monotonically when adding macroscopic state variables.

On the efficiency of the equation-free closure of statistical moments: dynamical properties of a stochastic epidemic model on Erdős–Rényi networks

We show how different explicit statistical moment closures, including the mean field and the Kirkwood approximations as well as an Ursell-type expansion for the moments, compare with the equation-free approach in the case of a stochastic epidemic model evolving on Erdős–Rényi networks. For illustration purposes we use a simple, discrete susceptible–infected–recovered stochastic model with a nonlinear recovering probability. For every closure scheme, we derive the corresponding macroscopic evolution equations and we construct the bifurcation diagrams with respect to the probability of infection. Finally, we construct the coarse-grained bifurcation diagram obtained with the equation-free method acting directly on the microscopic simulations, bypassing the derivation of explicit closures. We show that the equation-free approach captures the actual emergent nonlinear behavior and outperforms all the other explicit schemes.

1014 方程式フリー法を用いた一次元格子の力学解析(OS10. 電子・原子・マルチシミュレーションに基づく材料特性評価,OS11. 材料の組織・強度に関するマルチスケールアナリシス 合同ポスターセッション,オーガナイズドセッション講演)

1014 方程式フリー法を用いた一次元格子の力学解析(OS10. 電子・原子・マルチシミュレーションに基づく材料特性評価,OS11. 材料の組織・強度に関するマルチスケールアナリシス 合同ポスターセッション,オーガナイズドセッション講演)

An equation-free probabilistic steady-state approximation: Dynamic application to the stochastic simulation of biochemical reaction networks

Stochastic chemical kinetics more accurately describes the dynamics of "small" chemical systems, such as biological cells. Many real systems contain dynamical stiffness, which causes the exact stochastic simulation algorithm or other kinetic Monte Carlo methods to spend the majority of their time executing frequently occurring reaction events. Previous methods have successfully applied a type of probabilistic steady-state approximation by deriving an evolution equation, such as the chemical master equation, for the relaxed fast dynamics and using the solution of that equation to determine the slow dynamics. However, because the solution of the chemical master equation is limited to small, carefully selected, or linear reaction networks, an alternate equation-free method would be highly useful. We present a probabilistic steady-state approximation that separates the time scales of an arbitrary reaction network, detects the convergence of a marginal distribution to a quasi-steady-state, directly samples the underlying distribution, and uses those samples to accurately predict the state of the system, including the effects of the slow dynamics, at future times. The numerical method produces an accurate solution of both the fast and slow reaction dynamics while, for stiff systems, reducing the computational time by orders of magnitude. The developed theory makes no approximations on the shape or form of the underlying steady-state distribution and only assumes that it is ergodic. We demonstrate the accuracy and efficiency of the method using multiple interesting examples, including a highly nonlinear protein-protein interaction network. The developed theory may be applied to any type of kinetic Monte Carlo simulation to more efficiently simulate dynamically stiff systems, including existing exact, approximate, or hybrid stochastic simulation techniques.