Modeling Of Dynamic Systems Research Articles

Semi-supervised Learning of Dynamical Systems with Neural Ordinary Differential Equations: A Teacher-Student Model Approach

Modeling dynamical systems is crucial for a wide range of tasks, but it remains challenging due to complex nonlinear dynamics, limited observations, or lack of prior knowledge. Recently, data-driven approaches such as Neural Ordinary Differential Equations (NODE) have shown promising results by leveraging the expressive power of neural networks to model unknown dynamics. However, these approaches often suffer from limited labeled training data, leading to poor generalization and suboptimal predictions. On the other hand, semi-supervised algorithms can utilize abundant unlabeled data and have demonstrated good performance in classification and regression tasks. We propose TS-NODE, the first semi-supervised approach to modeling dynamical systems with NODE. TS-NODE explores cheaply generated synthetic pseudo rollouts to broaden exploration in the state space and to tackle the challenges brought by lack of ground-truth system data under a teacher-student model. TS-NODE employs an unified optimization framework that corrects the teacher model based on the student's feedback while mitigating the potential false system dynamics present in pseudo rollouts. TS-NODE demonstrates significant performance improvements over a baseline Neural ODE model on multiple dynamical system modeling tasks.

The necessity of the sausage-string structure for mode-locking regions of piecewise-linear maps

Piecewise-smooth maps are used as discrete-time models of dynamical systems whose evolution is governed by different equations under different conditions (e.g. switched control systems). By assigning a symbol to each region of phase space where the map is smooth, any period-p solution of the map can be associated to an itinerary of p symbols. As parameters of the map are varied, changes to this itinerary occur at border-collision bifurcations (BCBs) where one point of the periodic solution collides with a region boundary. It is well known that BCBs conform broadly to two cases: persistence, where the symbolic itinerary of a periodic solution changes by one symbol, and a nonsmooth-fold, where two solutions differing by one symbol collide and annihilate. This paper derives new properties of periodic solutions of piecewise-linear continuous maps on Rn to show that under mild conditions BCBs of mode-locked solutions on invariant circles must be nonsmooth-folds. This explains why Arnold tongues of piecewise-linear maps exhibit a sausage-string structure whereby changes to symbolic itineraries occur at codimension-two pinch points instead of codimension-one persistence-type BCBs. But the main result is based on the combinatorical properties of the itineraries, so the impossibility of persistence-type BCBs also holds when the periodic solution is unstable or there is no invariant circle.

Research on dynamic characteristics of railway side-cracked slab for train-track coupled system

The Ballastless track have been extensively implemented in high-speed railways. However, during the long-term operational process, degradation, such as side crack or defect phenomena produced on the slab inevitably, poses a threat to the stability and the durability of the track structure. Consequently, it is imperative to propose a dynamic approach that can effectively predict the impact of side-cracked slab (SCS) on the vehicle-track coupled system. This paper utilizes co-simulation technology to investigate the dynamic behavior of the slab with and without side crack. Firstly, the dynamic substructure of the finite element model of the SCS is obtained by eigenvalue analysis. Secondly, the SCS model has been applied to the classical vehicle-track coupled dynamic system, and the coupled dynamic model of the train-side cracked slab (TSCS) is established, which considers the multi-rigid-body vehicle subsystem, the flexible rail subsystem, the flexible SCS and subgrade foundation subsystem, and the wheel-rail nonlinear contact model. Later, the accuracy and reliability of the proposed model are verified by the comparison with field tests. Finally, the differences in the vibration characteristics of the vehicle and track structure with and without a side crack are compared and analyzed, and its dynamic stress of the slab as variable train speeds is revealed. The results indicate that the existence of a side crack will reduce the natural frequency of the slab, with a maximum decrease of 22.31%. Although the SCS has a minor effect on the vehicle responses, the existence of a side crack has a significant impact on the slab, especially on the root mean square (RMS) of the vibration acceleration of the slab, with a maximum difference of 74.10% laterally and an increase of 55.16% vertically. Meanwhile, the train speed remarkably affects the crack tip dynamic stress of the SCS, with a maximum increase of 76.07% in the longitudinal direction, 50.34% in the lateral direction and 49.09% in the vertical direction. Higher train speeds will further exacerbate crack propagation under long-term train loading. This study may contribute to the dynamic modeling of TSCS systems and the maintenance of the slab ballastless track.

Discover an accurate approximation of dynamical system without prior information and customized design

Modeling dynamical systems is fraught with challenges when data can be collected but thorough analysis of the mechanism is absent. We design a method to discover unknown dynamical systems from data. The method discovers an accurate approximation of the model without the prior information and the customized design for each problem. The identification steps are straightforward as bringing in the data and then obtaining the model. The method begins with the simple idea that the equations of motion of many practical problems are Riemann integrable functions. For this reason, the Fourier series can decompose the equations of motion. In order to improve the accuracy, we design an extension that helps us to approximate unknown functions by the Fourier series with a high rate of convergence. The idea converts the difficulty of modeling the dynamical system into finding its Fourier series approximation. Convenient procedures enable the modeling of different problems. Numerical examples show that the new method discovers linear and nonlinear dynamical systems in the same steps and without the prior information.

Preserving Lagrangian structure in data-driven reduced-order modeling of large-scale dynamical systems

This work presents a nonintrusive physics-preserving method to learn reduced-order models (ROMs) of Lagrangian systems, which includes nonlinear wave equations. Existing intrusive projection-based model reduction approaches construct structure-preserving Lagrangian ROMs by projecting the Euler–Lagrange equations of the full-order model (FOM) onto a linear subspace. This Galerkin projection step requires complete knowledge about the Lagrangian operators in the FOM and full access to manipulate the computer code. In contrast, the proposed Lagrangian operator inference approach embeds the mechanics into the operator inference framework to develop a data-driven model reduction method that preserves the underlying Lagrangian structure. The proposed approach exploits knowledge of the governing equations (but not their discretization) to define the form and parametrization of a Lagrangian ROM which can then be learned from projected snapshot data. The method does not require access to FOM operators or computer code. The numerical results demonstrate Lagrangian operator inference on an Euler–Bernoulli beam model, the sine-Gordon (nonlinear) wave equation, and a large-scale discretization of a soft robot fishtail with 779,232 degrees of freedom. The learned Lagrangian ROMs generalize well, as they can accurately predict the physical solutions both far outside the training time interval, as well as for unseen initial conditions.

Dietary mineral intakes predict Coronavirus-disease 2019 (COVID-19) incidence and hospitalization in older adults

BackgroundThe aim of this study was to determine the association between dietary mineral intake and Coronavirus-disease 2019 (COVID-19) infection and its associated hospitalization.MethodsThis cohort study utilized the MASHAD study population, which comprised individuals aged 35–65. Upon recruitment in 2007, dietary intake was documented using a validated 65-item food frequency questionnaire (FFQ). Data on COVID-19 PCR test results was collected from all relevant medical centers in Mashhad between February 2020 and June 2022. The regression model included dietary minerals and employed the backward variable selection method, along with advanced data analysis techniques.ResultsThe final analysis involved 1957 participants, including 193 COVID-19-positive patients. The mean age was 49.71 and 50.28 years in the COVID-19-positive and negative groups, respectively (p = 0.12). Dietary intakes of magnesium, iron, and potassium were notably lower in COVID-19-positive patients (P < 0.05). Following adjustments for age and sex, dietary iron remained significantly associated with COVID-19 incidence (OR = 0.94, 95% CI: 0.90–0.98). Furthermore, a statistically significant relationship was observed between dietary zinc and hospitalization due to COVID-19 (OR = 0.69, 95% CI: 0.51–0.93). In dynamical system models, intakes of calcium, zinc, and iron below the cut-offs of 1138, 9.7, and 8.17 mg/day, respectively, were linked to an increased risk of COVID-19 incidence.ConclusionHigher dietary iron and zinc intake are associated with decreased risk of COVID-19 infection and hospitalization, respectively.

Learning physics-based reduced-order models from data using nonlinear manifolds.

We present a novel method for learning reduced-order models of dynamical systems using nonlinear manifolds. First, we learn the manifold by identifying nonlinear structure in the data through a general representation learning problem. The proposed approach is driven by embeddings of low-order polynomial form. A projection onto the nonlinear manifold reveals the algebraic structure of the reduced-space system that governs the problem of interest. The matrix operators of the reduced-order model are then inferred from the data using operator inference. Numerical experiments on a number of nonlinear problems demonstrate the generalizability of the methodology and the increase in accuracy that can be obtained over reduced-order modeling methods that employ a linear subspace approximation.

Mathematical modeling of dynamic processes of agricultural mobile energy vehicles on an electric drive

The article considers the issues of modeling the processes of agricultural mobile energy vehicles (MEV) with electric drive (ED). A review of modern literature on the problem under study as well as questions on modeling the functional properties of mobile machines and improving the quality indicators of MEV work are presented. A mathematical model of the motion of the MEV with an electric motor is presented, as well as a description of the method used and a sequence of actions for conducting research. Preliminary theoretical studies of motion in different modes of operation have been carried out. The proposed model is convenient for implementation and calculation in any of the applied software products that support the modeling of dynamic systems with an electromechanical drive. The proposed model, the solution of which is based on the methods of numerical integration of systems in the Matlab Simulink software environment, made it possible to simulate the dynamic processes of electromechanical power transmission of MEV during various agricultural operations. With the help of this model, the analysis of electromechanical processes in transient and steady-state operating modes, as well as dynamic processes in the power transmission were carried out. Graphs of changes in the studied parameters of the MEV power transmission are obtained and elastic moments in the joints of the 5-mass design scheme are determined. The use of the model allows you to track the change in statistical characteristics when the conditions of the experiment change. The model has shown its operability when performing simulation of agricultural operations: fertilization, cultivation and sowing, and it can be used at the design stage to study the characteristics of dynamic processes of small-class traction MEV power transmissions with an electromechanical drive. With different parameters of the model, there is a change in the mathematical expectation of the angular velocity of the ED from 147.89 to 156.87 rad/s, a change in the mathematical expectation of the speed of the MES from 4.51 to 4.79 m/s.

AI Institute in Dynamic Systems: Developing machine learning and AI tools for scientific discovery, engineering design, and data‐driven control

AbstractThe mission of the AI Institute in Dynamic Systems is to develop the next generation of advanced machine learning (ML) and AI tools for controlling complex physical systems by discovering physically interpretable and physics‐constrained data‐driven models through optimal sensor selection and placement. The research effort is anchored by a common task framework (CTF) that evaluates the performance of ML algorithms, architectures, and optimization schemes for the diverse tasks required in engineering applications. The aim is to push beyond the boundaries of modern techniques by closing the loop between data collection, control, and modeling, creating a unique and cross‐disciplinary architecture for learning physically interpretable and physics constrained models of complex dynamic systems from time series data. The CTF further supports sustainable and open‐source challenge datasets, which are foundational for developing interpretable, ethical, and inclusive tools to solve problems fundamental to human safety, society, and the environment.

WyNDA: A method to discover mathematical models of dynamical systems from data

This paper introduces a novel method called Wide-Array of Nonlinear Dynamics Approximation (WyNDA) for extracting mathematical models of dynamical systems from data. A key advantage of this method over existing approaches lies in its suitability for online implementation. Moreover, WyNDA stands out by not relying on optimization or machine learning, ensuring computational efficiency. The fundamental concept revolves around approximating the unknown function of a dynamical system through a diverse set of basis functions that encapsulate the available data. An adaptive observer is then employed to iteratively refine this approximation and estimate the associated parameters. The efficacy of the proposed method is demonstrated through numerical simulations encompassing linear systems, nonlinear systems, and control systems. The results underscore the method’s ability to successfully unveil the governing equations of dynamical systems, highlighting its potential for extracting intricate system dynamics from observational data.•WyNDA represents a novel approach for uncovering mathematical models of dynamical systems from data.•Utilizing a series of basis functions, WyNDA effectively approximates the unknown structure inherent in dynamical systems.•The validation of WyNDA involves benchmark equations of dynamical systems, confirming its efficacy in diverse scenarios.

Waveformer for modeling dynamical systems

Neural operators have gained recognition as potent tools for learning solutions of a family of partial differential equations. The state-of-the-art neural operators excel at approximating the functional relationship between input functions and the solution space, potentially reducing computational costs and enabling real-time applications. However, they often fall short when tackling time-dependent problems, particularly in delivering accurate long-term predictions. In this work, we propose “waveformer”, a novel operator learning approach for learning solutions of dynamical systems. The proposed waveformer exploits wavelet transform to capture the spatial multi-scale behavior of the solution field and transformers for capturing the long horizon dynamics. We present four numerical examples involving Burgers’s equation, KS-equation, Allen–Cahn equation, and Navier–Stokes equation to illustrate the efficacy of the proposed approach. Results obtained indicate the capability of the proposed waveformer in learning the solution operator and show that the proposed Waveformer can learn the solution operator with high accuracy, outperforming existing state-of-the-art operator learning algorithms by up to an order, with its advantage particularly visible in the extrapolation region.

Research of nonstationary dynamic models of the organizational and technological process of railway transport technical vehicles repair

The problems of analytical and computer research of dynamic models of repair and operation of technical vehicles used in railway transport are among the urgent problems related to ensuring safety and stability of traffic, with the development of digital twins and the improvement of monitoring and diagnostic systems. The analysis of models described by various types of multidimensional nonlinear differential equations requires the use of modern methods of dynamical systems theory, numerical optimization and computer algebra. The purpose of the article is to conduct a qualitative and numerical analysis of mathematical models of organizational and technological processes (OTP) in railway transport, as well as the synthesis of new controlled OTP models. Nonstationary generalizations of a two-dimensional dynamic model describing the repair process of railway transport equipment are proposed. Various cases of using constant and time-dependent parameters are considered. The case of linear and exponential dependences of parameters on time is studied. For the studied models, trajectories are obtained taking into account the variation of constant parameters and taking into account the choice of parametric dependencies, a comparative analysis of the results is presented. The formulation of the problem of optimal control of the number of repaired and repaired railway wagons is proposed. Promising methods for solving this problem using numerical optimization methods, including those inspired by nature, are described. The results can be used in the tasks of constructing and analyzing mathematical models of dynamic systems, in the tasks of numerical modeling of transport systems and technological processes, in the tasks of optimizing the number of vehicles and improving the transportation process, as well as in the tasks of creating digital counterparts in the transport industry.

Possibilities and limits of modelling of long-range economic consequences of air pollution – A case study

Air pollution is the biggest environmental problem in modern societies, causing considerable health damage and requiring substantial financial resources for health care. The goal of the study is to demonstrate the adverse economic consequences of air pollution on example of a small, open Central European country, Hungary, and to provide quantified financial arguments for macroeconomic decision-making for the development of a long-term energy strategy. On the basis of the Cobb-Douglas production function and Solow-Swann model of dynamic economic systems a simple and robust model was constructed to estimate and predict economic losses, caused by the pollution. On base of results it is obvious, that on base of macroeconomic theory and combination of various, publicly available, quality-controlled statistical resources quantifiable models can be constructed to characterise the economic consequences of air pollution, but it should be taken into consideration, that the reliability of economic models considerably depends on their initial parameters and practical validity of assumptions, based on which the underlying economic theories were built. The most important economic burden of air pollution is caused by the loss of working–age population, resulting in a decrease of 4.1–9.4 % a year in Gross Domestic Product (GDP) in the next fifty years. The additional burden of health care costs amounts to 0.1 % of GDP. Reducing air pollution is not only a quality of life improvement but also an investment into the economic development. Notwithstanding of statistical biases it could be proven the importance of combination health economic and econometric methods in preparation of more efficient environmental-related socio-economic decisions.

Nonlinear causal network learning via Granger causality based on extreme support vector regression.

For complex networked systems, based on the consideration of nonlinearity and causality, a novel general method of nonlinear causal network learning, termed extreme support vector regression Granger causality (ESVRGC), is proposed. The nonuniform time-delayed influence of the driving nodes on the target node is particularly considered. Then, the restricted model and the unrestricted model of Granger causality are, respectively, formulated based on extreme support vector regression, which uses the selected time-delayed components of system variables as the inputs of kernel functions. The nonlinear conditional Granger causality index is finally calculated to confirm the strength of a causal interaction. Generally, based on the simulation of a nonlinear vector autoregressive model and nonlinear discrete time-delayed dynamic systems, ESVRGC demonstrates better performance than other popular methods. Also, the validity and robustness of ESVRGC are also verified by the different cases of network types, sample sizes, noise intensities, and coupling strengths. Finally, the superiority of ESVRGC is successful verified by the experimental study on real benchmark datasets.

Some methodological aspects of mathematical modeling in dynamic systems

This paper is devoted to methodologies in the field of mathematical foundations of information technology and computer sciences. The theoretical foundations of mathematical modeling of dynamic systems are briefly presented. Various aspects of the concept of a model are considered. The division into types of models is presented. The steps of the mathematical modeling process are detailed. Basic requirements for mathematical models are formulated. The main approaches used in the process of mathematical modeling are described.

Mathematical modelling with computational fractional order for the unfolding dynamics of the communicable diseases

Mathematical models based on computational fractional orders, employed for accurate modelling of complex dynamic systems, can ensure the implementation of various analytical, numerical and computing methods encompassing their applications to emerging and ever-varying real-world problems. Tracking, managing and controlling communicable diseases, one being monkeypox with different features, virological and taxonomic attributes, are oriented towards high-risk groups concerning global public health. This study, accordingly, is devoted to the presentation of the piecewise global derivative model of the monkeypox virus by applying the Caputo and Atangana Baleanu fractional-order derivatives in the partitioned two sub-intervals. The model includes eight compartments with two categories of human and rodent populations. The cases which take part in some sense for the said infection are investigated along with connection in this format. The existence and uniqueness of the solution in the framework of the piecewise global derivative are analyzed for both sub-intervals using fixed point theory. The detailed investigation of the dynamics of fractional-order systems and among many other dynamic features, stability is addressed. The stability of the solution is, thus, examined using the idea of Ulam Hyers concept. For the best fitting values of the parameters, the results are simulated using the monkeypox data. Using the method of Newton polynomial, different piecewise dynamics of each compartment are simulated on different fractional orders and time durations. This kind of a proposed approach is thought to lay a foundation where the transmission takes place to control epidemic events and other infectious medical conditions through vaccines or taking preventive measures to maintain and advance global public health while fully optimizing the clinical care of the diseases to manage complications, alleviate symptoms as well as prevent the long-term sequelae. This analysis also deals with sudden variation in monkeypox dynamics and also for crossover dynamics along with removal of discontinuity through modification of piecewise global analysis.

Data-driven analysis and prediction of dynamic postprandial metabolic response to multiple dietary challenges using dynamic mode decomposition.

In the field of precision nutrition, predicting metabolic response to diet and identifying groups of differential responders are two highly desirable steps toward developing tailored dietary strategies. However, data analysis tools are currently lacking, especially for complex settings such as crossover studies with repeated measures.Current methods of analysis often rely on matrix or tensor decompositions, which are well suited for identifying differential responders but lacking in predictive power, or on dynamical systems modeling, which may be used for prediction but typically requires detailed mechanistic knowledge of the system under study. To remedy these shortcomings, we explored dynamic mode decomposition (DMD), which is a recent, data-driven method for deriving low-rank linear dynamical systems from high dimensional data.Combining the two recent developments "parametric DMD" (pDMD) and "DMD with control" (DMDc) enabled us to (i) integrate multiple dietary challenges, (ii) predict the dynamic response in all measured metabolites to new diets from only the metabolite baseline and dietary input, and (iii) identify inter-individual metabolic differences, i.e., metabotypes. To our knowledge, this is the first time DMD has been applied to analyze time-resolved metabolomics data. We demonstrate the potential of pDMDc in a crossover study setting. We could predict the metabolite response to unseen dietary exposures on both measured (R2 = 0.40) and simulated data of increasing size (= 0.65), as well as recover clusters of dynamic metabolite responses. We conclude that this method has potential for applications in personalized nutrition and could be useful in guiding metabolite response to target levels. The measured data analyzed in this study can be provided upon reasonable request. The simulated data along with a MATLAB implementation of pDMDc is available at https://github.com/FraunhoferChalmersCentre/pDMDc.

Koopman-based surrogate models for multi-objective optimization of agent-based systems

Agent-based models (ABMs) provide an intuitive and powerful framework for studying social dynamics by modeling the interactions of individuals from the perspective of each individual. In addition to simulating and forecasting the dynamics of ABMs, the demand to solve optimization problems to support, for example, decision-making processes naturally arises. Most ABMs, however, are non-deterministic, high-dimensional dynamical systems, so objectives defined in terms of their behavior are computationally expensive. In particular, if the number of agents is large, evaluating the objective functions often becomes prohibitively time-consuming. We consider data-driven reduced models based on the Koopman generator to enable the efficient solution of multi-objective optimization problems involving ABMs. In a first step, we show how to obtain data-driven reduced models of non-deterministic dynamical systems (such as ABMs) that depend potentially nonlinearly on control inputs. We then use them in the second step as surrogate models to solve multi-objective optimal control problems. We first illustrate our approach using the example of a voter model, where we compute optimal controls to steer the agents to a predetermined majority, and then using the example of an epidemic ABM, where we compute optimal containment strategies in a prototypical situation. We demonstrate that the surrogate models effectively approximate the Pareto-optimal points of the ABM dynamics by comparing the surrogate-based results with test points, where the objectives are evaluated using the ABM. Our results show that when objectives are defined by the dynamic behavior of ABMs, data-driven surrogate models support or even enable the solution of multi-objective optimization problems.

Data-Driven Modeling of the Nonlinear Dynamics of Passive Lower-Limb Prosthetic Systems

Abstract Modeling the nonlinear dynamics of prosthetic feet is an important tool for linking prosthesis mechanical properties to end-user outcomes. There has been a renewed interest in data-driven modeling of dynamical systems, with the development of the Extended Dynamic Mode Decomposition with control (eDMDc) and the Sparse Identification of Nonlinear Dynamics with Control (SINDYc). These algorithms do not require prior information about the system, including mechanical configuration, and are data-driven. The aim of this study was to assess the feasibility and accuracy of applying these data-driven algorithms to model prosthesis nonlinear load response dynamics. Different combinations of a dynamic response foot, a hydraulic ankle unit, and three shock-absorbing pylons of varying resistance were tested loaded and unloaded at three orientations reflecting critical positions during the stance phase of walking. We tested two different data-driven algorithms, the eDMDc, with two different kernels, and the SINDYc, which regresses the coefficients for a nonlinear ordinary differential equation. Each algorithm was able to model the nonlinear prosthesis dynamics, but the SINDYc outperformed the eDMDc methods with a root mean square error across orientations &amp;lt; 1.50 mm and a maximum error in peak displacement of 1.28 mm or 4% relative error. From the estimated SINDYc governing equation of the system dynamics, we were able to simulate different mechanical behavior by systematically varying parameter values, which offers a novel foundation for designing, controlling, and classifying prosthetic systems ultimately aimed at improving prosthesis user outcomes.

Volterra Black-Box Models Identification Methods: Direct Collocation vs. Least Squares

The Volterra integral-functional series is the classic approach for nonlinear black box dynamical system modeling. It is widely employed in many domains including radiophysics, aerodynamics, electronic and electrical engineering and many others. Identifying the time-varying functional parameters, also known as Volterra kernels, poses a difficulty due to the curse of dimensionality. This refers to the exponential growth in the number of model parameters as the complexity of the input-output response increases. The least squares method (LSM) is widely acknowledged as the standard approach for tackling the issue of identifying parameters. Unfortunately, the LSM suffers with many drawbacks such as the sensitivity to outliers causing biased estimation, multicollinearity, overfitting and inefficiency with large datasets. This paper presents an alternative approach based on direct estimation of the Volterra kernels using the collocation method. Two model examples are studied. It is found that the collocation method presents a promising alternative for optimization, surpassing the traditional least squares method when it comes to the Volterra kernels identification including the case when input and output signals suffer from considerable measurement errors.