Modeling Of Complex Dynamic Systems Research Articles

A Compositional Semantics of Boolean-Logic Driven Markov Processes

Boolean-logic driven Markov processes (BDMPs) is a prominent dynamic extension of static fault trees to model repairable and complex dynamic systems. While BDMPs are intensively used in an industrial context for dependability analysis of energy systems, its formal semantics has not been systematically treated. To date, BDMPs are defined as a library of the domain-specific dependability-modelling language Figaro, a library that is neither open source nor publicly available. A rigorous semantic underpinning of BDMPs is indispensable for (1) developing BDMP analysis tools and (2) comparing its expressive power to other related reliability modelling languages. This paper presents a formal semantics to BDMPs using Markov automata (MA), an extension of continuous-time Markov chains (CTMCs) with action transitions that can be used to compose complex MA from smaller MA. This enables us to provide a compositional semantics. That is, we express the semantics of each individual BDMP element as an MA and obtain the MA for the entire BDMP by combining the MA of its elements. This makes the semantics comprehensible, for those who are familiar with automata theory, and easily extensible with new BDMP elements, e.g., to model security aspects. After the entire BDMP is considered, the actions in its MA that were used to “glue” the MA of BDMP elements, are ignored. This results in a CTMC that is amenable to exact numerical analysis by, e.g., efficient probabilistic model-checking techniques. We report on a prototypical implementation of our semantics and empirically show that our semantics yields dependability metrics that correspond to the interpretation by the Figaro knowledge base of BDMPs.

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.

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.

Optimizing Physics-Informed Neural Network in Dynamic System Simulation and Learning of Parameters

Artificial neural networks have changed many fields by giving scientists a strong way to model complex phenomena. They are also becoming increasingly useful for solving various difficult scientific problems. Still, people keep trying to find faster and more accurate ways to simulate dynamic systems. This research explores the transformative capabilities of physics-informed neural networks, a specialized subset of artificial neural networks, in modeling complex dynamical systems with enhanced speed and accuracy. These networks incorporate known physical laws into the learning process, ensuring predictions remain consistent with fundamental principles, which is crucial when dealing with scientific phenomena. This study focuses on optimizing the application of this specialized network for simultaneous system dynamics simulations and learning time-varying parameters, particularly when the number of unknowns in the system matches the number of undetermined parameters. Additionally, we explore scenarios with a mismatch between parameters and equations, optimizing network architecture to enhance convergence speed, computational efficiency, and accuracy in learning the time-varying parameter. Our approach enhances the algorithm’s performance and accuracy, ensuring optimal use of computational resources and yielding more precise results. Extensive experiments are conducted on four different dynamical systems: first-order irreversible chain reactions, biomass transfer, the Brusselsator model, and the Lotka-Volterra model, using synthetically generated data to validate our approach. Additionally, we apply our method to the susceptible-infected-recovered model, utilizing real-world COVID-19 data to learn the time-varying parameters of the pandemic’s spread. A comprehensive comparison between the performance of our approach and fully connected deep neural networks is presented, evaluating both accuracy and computational efficiency in parameter identification and system dynamics capture. The results demonstrate that the physics-informed neural networks outperform fully connected deep neural networks in performance, especially with increased network depth, making them ideal for real-time complex system modeling. This underscores the physics-informed neural network’s effectiveness in scientific modeling in scenarios with balanced unknowns and parameters. Furthermore, it provides a fast, accurate, and efficient alternative for analyzing dynamic systems.

Learning skillful medium-range global weather forecasting.

Global medium-range weather forecasting is critical to decision-making across many social and economic domains. Traditional numerical weather prediction uses increased compute resources to improve forecast accuracy but does not directly use historical weather data to improve the underlying model. Here, we introduce GraphCast, a machine learning-based method trained directly from reanalysis data. It predicts hundreds of weather variables for the next 10 days at 0.25° resolution globally in under 1 minute. GraphCast significantly outperforms the most accurate operational deterministic systems on 90% of 1380 verification targets, and its forecasts support better severe event prediction, including tropical cyclone tracking, atmospheric rivers, and extreme temperatures. GraphCast is a key advance in accurate and efficient weather forecasting and helps realize the promise of machine learning for modeling complex dynamical systems.

An Improved Approach for Implementing Dynamic Mode Decomposition with Control

Dynamic Mode Decomposition with Control is a powerful technique for analyzing and modeling complex dynamical systems under the influence of external control inputs. In this paper, we propose a novel approach to implement this technique that offers computational advantages over the existing method. The proposed scheme uses singular value decomposition of a lower order matrix and requires fewer matrix multiplications when determining corresponding approximation matrices. Moreover, the matrix of dynamic modes also has a simpler structure than the corresponding matrix in the standard approach. To demonstrate the efficacy of the proposed implementation, we applied it to a diverse set of numerical examples. The algorithm’s flexibility is demonstrated in tests: accurate modeling of ecological systems like Lotka-Volterra, successful control of chaotic behavior in the Lorenz system and efficient handling of large-scale stable linear systems. This showcased its versatility and efficacy across different dynamical systems.

Probabilities Are Not Enough: Formal Controller Synthesis for Stochastic Dynamical Models with Epistemic Uncertainty

Capturing uncertainty in models of complex dynamical systems is crucial to designing safe controllers. Stochastic noise causes aleatoric uncertainty, whereas imprecise knowledge of model parameters leads to epistemic uncertainty. Several approaches use formal abstractions to synthesize policies that satisfy temporal specifications related to safety and reachability. However, the underlying models exclusively capture aleatoric but not epistemic uncertainty, and thus require that model parameters are known precisely. Our contribution to overcoming this restriction is a novel abstraction-based controller synthesis method for continuous-state models with stochastic noise and uncertain parameters. By sampling techniques and robust analysis, we capture both aleatoric and epistemic uncertainty, with a user-specified confidence level, in the transition probability intervals of a so-called interval Markov decision process (iMDP). We synthesize an optimal policy on this iMDP, which translates (with the specified confidence level) to a feedback controller for the continuous model with the same performance guarantees. Our experimental benchmarks confirm that accounting for epistemic uncertainty leads to controllers that are more robust against variations in parameter values.

Extension of Dynamic Mode Decomposition for dynamic systems with incomplete information based on t-model of optimal prediction

The Dynamic Mode Decomposition has proved to be a very efficient technique to study dynamic data. This is entirely a data-driven approach that extracts all necessary information from data snapshots which are commonly supposed to be sampled from measurement. The application of this approach becomes problematic if the available data is incomplete because some dimensions of smaller scale either missing or unmeasured. Such setting occurs very often in modeling complex dynamical systems such as power grids, in particular with reduced-order modeling. To take into account the effect of unresolved variables, the optimal prediction approach based on the Mori-Zwanzig formalism can be applied to obtain the most expected prediction under existing uncertainties. This effectively leads to the development of a time-predictive model accounting for the impact of missing data. In the present paper we provide a detailed derivation of the considered method from the Liouville equation and finalize it with the optimization problem that defines the optimal transition operator corresponding to the observed data. In contrast to the existing approach, we consider a first-order approximation of the Mori-Zwanzig decomposition, state the corresponding optimization problem and solve it with the gradient-based optimization method. The gradient of the obtained objective function is computed precisely through the automatic differentiation technique. The numerical experiments illustrate that the considered approach gives practically the same dynamics as the exact Mori-Zwanzig decomposition, but is less computationally intensive.

Mathematical modelling of oscillations of a machine for cutting tops of root crops

High-quality cutting of root crop tops at the root ensures both a high yield and determines the condition of the root crops before they are extracted from the soil. Therefore, the search for conditions that will ensure this is an important and urgent problem for the production of these crops. The purpose of this research was to determine, through an analytical study, the optimal parameters of a new design of a machine for harvesting root crop tops, which is mounted in front of an aggregating tractor and performs oscillatory movements in the longitudinal-vertical plane. Methods related to the modelling of complex dynamic systems consisting of an aggregating tractor and a front-mounted machine, the development and solution of the final form of differential equations of oscillatory motion and computer modelling of the oscillatory process were used. According to the developed equivalent scheme, a new differential equation of angular vibrations of the machine was solved in the final form. In addition, numerical modelling was performed on a PC, which allowed constructing graphical dependencies of the machine’s turning angle φ at different speeds V of its forward movement and values of the c stiffness coefficient and μ damping coefficient of the pneumatic tyres of the copying wheels, and at different values of the unevenness of the soil surface and the main design parameters of the machine. It was established that with an increase V from 1.5 m·s-1 to 2.5 m·s-1, the amplitude of oscillations of the machine’s turning angle φ increased from 0.88° to 1.18°. However, at V 1.5 m·s-1, the duration of the transient process is 0.22 s, and at a speed of 2.5 m·s-1, this figure is already 0.14 s, i.e., a decrease of 36%. The positive amplitude of oscillations φ of the machine’s steering angle reaches 1.2°, and the negative amplitude does not exceed 0.3°, i.e., the oscillation range is insignificant. In the range of values considered V, preference should be assigned to its higher value. The structural and kinematic parameters of the system examined were determined using computer simulation. The presented method of mathematical modelling of the oscillatory process can be used in the research of any machines that are hitched in front of the aggregating tractors

Application of Physics-Informed Neural Networks Algorithm to Predict the Vorticity of a Moving Cylindrical Flow Field

Turbulence is a typical physical phenomenon which is involved in many engineering fields. The combination of machine learning and turbulence modeling is an emerging research direction in the field of fluid mechanics. The current achievements in this research direction have strongly verified its feasibility and indicated a positive prospect for the application of machine learning for the turbulence modeling. Machine learning can help discover models of complex dynamical systems from the data directly. In this work, we apply the machine learning algorithm called the physics-informed neural networks (PINNs) to predict the vorticity of a moving cylindrical flow field. Through the neural network method based on physical information, a neural network model is established to simulate the flow around a moving cylinder. Results demonstrate that the vorticity predicted by PINNs algorithm are in good agreement with the benchmark results.

Surrogate models of complex dynamic systems from energy distributions

Engineers who design complex dynamic systems do not have the luxury of iteration. Numerical analysis of a single design, typically by finite element analysis, requires enormous amounts of CPU time. Therefore, improving a design by trying out many design modifications is impossible. Moreover, it is also impossible to identify which subsystem designs have the largest effects on the dynamic response of the system. The present work addresses this challenge by constructing surrogate models from energy distributions in a nominal design. These surrogate models are surprisingly effective at identifying subsystems that have the largest effects on the dynamic response, and therefore guide engineers to design modifications that matter most. A significant advantage of these surrogate models is the very low cost of creating them, as they are created directly from the frequency response of a nominal design and therefore do not require additional linear solves or other time-intensive calculations. This presentation begins with an analytical justification for considering energy distributions as surrogate models. Next, Monte Carlo Simulations are presented to show strong correlations between energy distributions and the effects of design changes on dynamic responses. These simulations include random structures as well as beams. Work supported by ONR under Grant N00014-19-1-2100.

Structural damage assessment through a new generalized autoencoder with features in the quefrency domain

Recently, the challenges in modeling complex dynamical systems, and the advancement in machine learning methodologies have indicated a new promising direction for damage assessment in civil and mechanical systems. Powerful and efficient data-driven approaches have been increasingly employed in Structural Health Monitoring (SHM) to extract Damage Sensitive Features (DSFs) from the monitored dynamic response of structures. In this study, a New Generalized Auto-Encoder (NGAE), integrated with a statistical-pattern-recognition-based approach that uses the power cepstral coefficients of structural acceleration responses as DSFs, is proposed for structural damage assessment. This NGAE is well-generalized in the components of cepstral coefficients that represent the structural properties of the entire system thanks to a newly defined encoder-decoder mapping, which largely reduces rid of the data variance attributed to different types of excitations and measurement noise. The cepstral coefficients, by virtue of a compact representation of the structural properties, can greatly simplify the structure of the NGAE, and therefore, significantly accelerate training and inference speeds with very few computational requirements. Two specific evaluation metrics that relates to the autoencoder signal reconstruction error are defined and used to assess the presence of damage. The proposed method has been validated through numerical simulations and experimental data, and shows better performance compared to a Traditional Auto-Encoder (TAE) and the Principal Component Analysis (PCA).

The Fractal Tapestry of Life: II Entailment of Fractional Oncology by Physiology Networks.

This is an essay advocating the efficacy of using the (noninteger) fractional calculus (FC) for the modeling of complex dynamical systems, specifically those pertaining to biomedical phenomena in general and oncological phenomena in particular. Herein we describe how the integer calculus (IC) is often incapable of describing what were historically thought to be simple linear phenomena such as Newton's law of cooling and Brownian motion. We demonstrate that even linear dynamical systems may be more accurately described by fractional rate equations (FREs) when the experimental datasets are inconsistent with models based on the IC. The Network Effect is introduced to explain how the collective dynamics of a complex network can transform a many-body noninear dynamical system modeled using the IC into a set of independent single-body fractional stochastic rate equations (FSREs). Note that this is not a mathematics paper, but rather a discussion focusing on the kinds of phenomena that have historically been approximately and improperly modeled using the IC and how a FC replacement of the model better explains the experimental results. This may be due to hidden effects that were not anticapated in the IC model, or to an effect that was acknowledged as possibly significant, but beyond the mathematical skills of the investigator to Incorporate into the original model. Whatever the reason we introduce the FRE used to describe mathematical oncology (MO) and review the quality of fit of such models to tumor growth data. The analytic results entailed in MO using ordinary diffusion as well as fractional diffusion are also briefly discussed. A connection is made between a time-dependent fractional-order derivative, technically called a distributed-order parameter, and the multifractality of time series, such that an observed multifractal time series can be modeled using a FRE with a distributed fractional-order derivative. This equivalence between multifractality and distributed fractional derivatives has not received the recognition in the applications literature we believe it warrants.

Adaptive Semiparametric Bayesian Differential Equations Via Sequential Monte Carlo

Nonlinear differential equations (DEs) are used in a wide range of scientific problems to model complex dynamic systems. The differential equations often contain unknown parameters that are of scientific interest, which have to be estimated from noisy measurements of the dynamic system. Generally, there is no closed-form solution for nonlinear DEs, and the likelihood surface for the parameter of interest is multi-modal and very sensitive to different parameter values. We propose a Bayesian framework for nonlinear DE systems. A flexible nonparametric function is used to represent the dynamic process such that expensive numerical solvers can be avoided. A sequential Monte Carlo algorithm in the annealing framework is proposed to conduct Bayesian inference for parameters in DEs. In our numerical experiments, we use examples of ordinary differential equations and delay differential equations to demonstrate the effectiveness of the proposed algorithm. We developed an R package that is available at https://github.com/shijiaw/smcDE. Supplementary files for this article are available online.

Vibration monitoring of the state of preservation of historic buildings

architectural monument. A technical inspection of the building was performed and existing defects were identified. The analysis of potential sources of dynamic influences on the building is carried out.
 The technique of experimental researches is developed. The technique is based on a systems approach and involves the determination of dynamic parameters with the location of sensors directly on the individual elements of the studied structures. During the study, records of continuous fixation of the distribution of active oscillations of the structure were used.
 Numerical values of oscillation frequencies at different external sources are obtained. The nature of the distribution of amplitude values of oscillations in structures testified to the integrity of the structure as a single dynamic system.
 A comprehensive approach to solving the problem of modeling complex dynamical systems is considered.
 A mathematical model of a fragment of structures has been developed. The elements of the model create a multilayer structure with variable elastic characteristics. The results of numerical studies of the behavior of structures, taking into account the presence of peeling plaster, indicate a significant increase in natural frequencies of oscillation (up to 10 times), taking into account the partial lack of adhesion between plaster and load-bearing structures. As a result of separation, the plaster begins to work as a separate element with its own frequency with which it can resonate with other vibrating sources.
 The method of experimental-numerical analysis of the object is offered. This approach is based on the premise of determining the dynamic parameters of the interaction of subsystems with each other. The obtained numerical values of dynamic parameters will be used in further monitoring of the investigated building, on the basis of which it is possible to predict possible scenarios of behavior of structures over time.
 The research results can be used in the restoration process for optimal fixing of plaster fragments and avoidance of resonant phenomena in operation on the basis of detailed local mathematical models.

Dynamic Optimization Algorithm Based on Multi-modal Niche Lion Swarm by Sensitive Individuals

Aiming at the problem that the model of complex dynamic system which is difficult to track its global optimal solution, this paper proposes a multi-modal niche lion swarm optimization algorithm based on sensitive individuals to solve the optimization problem of complex dynamic system. Sensitive individuals are set in the lion swarm algorithm to perceive changes in the dynamic environment, and to track the global optimal value in time. It combines with niche technology to increase the diversity of the population and avoids falling into local extremes. The dynamic environment is simulated by the dynamic function generator and compared with other optimization algorithm to verify the superiority of the algorithm proposed in this paper.

Extended dynamic mode decomposition for inhomogeneous problems

Dynamic mode decomposition (DMD) is a powerful data-driven technique for construction of reduced-order models of complex dynamical systems. Multiple numerical tests have demonstrated the accuracy and efficiency of DMD, but mostly for systems described by homogeneous partial differential equations (PDEs) with homogeneous boundary conditions. We propose an extended dynamic mode decomposition (xDMD) approach to cope with the potential unknown sources/sinks in PDEs. Motivated by similar ideas in deep neural networks, we equip our xDMD with two new features. First, it has a bias term, which accounts for inhomogeneity of PDEs and/or boundary conditions. Second, instead of learning a flow map, xDMD learns the residual increment by subtracting the identity operator. Our theoretical error analysis demonstrates the improved accuracy of xDMD relative to standard DMD. Several numerical examples are presented to illustrate this result.

Searching for Errors in Models of Complex Dynamic Systems.

Mathematical modeling is seen as a key step to understand, predict, and control the temporal dynamics of interacting systems in such diverse areas like physics, biology, medicine, and economics. However, for large and complex systems we usually have only partial knowledge about the network, the coupling functions, and the interactions with the environment governing the dynamic behavior. This incomplete knowledge induces structural model errors which can in turn be the cause of erroneous model predictions or misguided interpretations. Uncovering the location of such structural model errors in large networks can be a daunting task for a modeler. Here, we present a data driven method to search for structural model errors and to confine their position in large and complex dynamic networks. We introduce a coherence measure for pairs of network nodes, which indicates, how difficult it is to distinguish these nodes as sources of an error. By clustering network nodes into coherence groups and inferring the cluster inputs we can decide, which cluster is affected by an error. We demonstrate the utility of our method for the C. elegans neural network, for a signal transduction model for UV-B light induced morphogenesis and for synthetic examples.

Designing optimal models of nonlinear MIMO systems based on orthogonal polynomial neural networks

ABSTRACT This paper presents a new method for modelling of dynamic systems by using specially designed orthogonal polynomial neural networks. These networks utilize the feature that the basis made of orthogonal functions can be used for approximation of arbitrary function, while their property of orthogonality enables optimal performances in the sense of both convergence time and approximation error. In this regard, generalized quasi-orthogonal polynomials, specifically tailored for the application in the modelling of complex dynamic systems with time-varying behaviour, are considered. Adaptivity of the designed model is achieved by using variable factors inside the orthogonal basis. The designed orthogonal neural network is applied in modelling of laboratory twin-rotor aero-dynamic system as a representative of nonlinear multiple input-multiple output systems. Detailed comparative analysis is performed for a different number of polynomials in expansion with the purpose of finding the optimal model in the sense of trade-off between model accuracy and complexity.

Gas turbine monitoring using neural network dynamic nonlinear autoregressive with external exogenous input modelling

The main purpose of the present work is to propose an effective tool which allows to ensure the protection and the safety measures against the instability phenomena in a gas turbine based on the modelling of its dynamic behaviour. In order to provide an efficient diagnostic strategy for this type of rotating machine, a supervision system based on the development of artificial neural network tools is proposed in this paper. Where, the dynamic nonlinear autoregressive approach with external exogenous input NARX is used for the identification of the studied system dynamics, to monitor the vibrational dynamics of the operating turbine. This leads to establishing a solution for the different ranges of rotational speed and ensuring dynamic stability through the vibration indicators, determined by the proposed neural network approach. Also, offer a normalized mean square error on the order of 3.8414e−3 for the high-pressure turbine, 1.29152e−1 for the gas control valve and 2.12090 e-4 for the air control valve. Furthermore, it permits the vibration monitoring and efficiently extracts the essentials of dynamic model behaviour, to effectively size the operating gas turbine system.The obtained results of the application of the proposed approach on the gas turbine system presented in this paper proves its ability for the detection and the management on real-time of the eventual failures caused mainly by intrinsic vibrations. On the other side, these results prove clearly the effectiveness of the use of the artificial neural networks as a very powerful calculation tools in the modelling of complex dynamic systems.