The network takeover

This article introduces artificial intelligence techniques in mathematical modelling of complex systems and their applications. Mathematical modelling of complex systems is a method of studying the structure and behaviour of complex systems, aiming to understand interactions and nonlinear effects in the system. Commonly used modelling methods include system dynamics, network theory, and algebraic methods. Artificial intelligence technologies include machine learning and deep learning, which can be used for tasks such as prediction and classification, anomaly detection, optimization and decision-making. In mathematical modelling of complex systems, artificial intelligence technology can learn system patterns and laws from large amounts of data, and can be applied to image and speech recognition, time series analysis and other fields. Deep learning and machine learning are important branches of artificial intelligence. They realize the modelling and analysis of complex systems by building neural network models. Data-driven modelling is a modelling method based on actual data that, combined with traditional theoretical modelling, can better describe and predict the behaviour of complex systems. Self-control of complex systems means that the system realizes its own optimization and adjustment through adaptive control algorithms and feedback mechanisms. In summary, artificial intelligence technology has broad application prospects in mathematical modelling of complex systems and will provide new tools and methods for in-depth understanding and solving problems in complex systems.

This paper investigates how to include artificial intelligence (AI) methods into complex system mathematical modelling. Traditionally, techniques like system dynamics, network theory, and algebraic modelling are used to study complex systems, which are distinguished by dynamic interactions, nonlinear behaviour, and high dimensionality. The emergence of artificial intelligence (AI), namely machine learning (ML) and deep learning (DL), has created new possibilities for improving forecast accuracy, revealing latent patterns, and allowing adaptive behaviour in these systems. Because AI algorithms are so good at learning from massive amounts of data, they may be used for time series forecasting, anomaly detection, optimisation, and decision-making. In fields like image and speech recognition, bioinformatics, and autonomous systems, neural network models—which are essential to machine learning and deep learning—have shown an amazing capacity to represent and analyse complicated phenomena. Additionally, the integration of data-driven modelling with conventional theoretical frameworks improves the capacity to capture system behaviours that are challenging to represent analytically. Complex system optimisation and self-regulation are further supported by AI-enabled adaptive control systems. In conclusion, combining AI and mathematical modelling enhances simulation accuracy while offering a revolutionary toolkit for comprehending and controlling complex systems in the fields of science, industry, and society. Keyword: Adaptive control of complex systems, data-driven modelling, machine learning and deep learning, mathematical modelling of complex systems, and artificial intelligence technology.

Precise dynamic mathematical models of complex systems are important in control and diagnostic systems design and allow testing a complex system in virtual environment at a low cost. They can be also utilized in rapid prototyping using a concept of hardware in the loop. Ever improving methods of experimental identification and using approaches in non-linear approximation can considerably increase the precision of dynamic models of complex systems. The article deals with non-linear approximation of transfer gains of a complex system and evaluates the influence of operational point selection on precision of the resulting model using methods of experimental data driven identification. The object of control is represented by two similar small turbojet engines at the Departments of the authors, the iSTC-21v and TKT-1, both based on the same power section having two degrees of freedom: fuel mass flow rate and variable convergent nozzle position.

The workshop Reactive Flow and Transport Through Complex Systems , organized by Cornelius J. van Duijn (Eindhoven), Andro Mikelić (Lyon) and Christoph Schwab (Zürich) was held October 30th–November 5th, 2005. This meeting was attended by over 46 participants with broad geographic representation from all continents.The theme of the conference, modeling, analysis and numerical simulation of diffusion and transport processes in complex systems , is a response to the need for more accurate, quantitative prediction in a growing number of scientific disciplines, particularly those related to biological applications. Here, simple mathematical models have been found, in particular due to the vastly increased available experimental data from these systems, to offer only inadequate and incomplete understanding of the observed phenomena. This resulted in increased requirements for quantitative, verified predictions from sophisticated mathematical as well as computational models. The continuous development of complex mathematical and computational models and their verification and validation against available experimental data is a continuous source of challenges for applied and computational mathematicians. The complexity of the systems arises from several sources: highly irregular geometries of membranes and interfaces (as, e.g. in bone marrow, cell membranes, root systems of plants, membrane structures in human organs), physical or chemical properties of the systems (e.g., models for spread of pollution in underground medium which has uncertain material properties, where chemical reactions take place between constituents, and where strong transport effects on a macroscopic scale coexist with diffusion phenomena at the grain interfaces). Quantitative mathematical and computational models of such phenomena are not only essential for a deeper understanding of these systems but, at least equally importantly, are a keystone in the development of new technologies which increasingly mimic and adapt biological phenomena for industrial purposes (e.g., root-reactor technology for the efficient production of organic compounds, bioinspired catalysts for waste processing, to name but a few). Accordingly, the rather wide scope of the topic of the conference and the blend of researchers working in several areas of applied mathematics was a necessary condition to review modelling approaches across a number of application areas as well as across several mathematical disciplines. Accordingly, during the meeting, talks were presented on homogenization, analysis and computation of multiscale problems, models of porous media, biological flow problems, to name but a few. In addition to the regular presentations, there were three evening sessions organized “on the spot” based on the discussions which started in the first half of the meeting. These were in each case opened by a presentation from a person invited by the organizers, and were devoted to the topics: The presentations of the experts present at the meeting comprised, naturally, a much wider scope of topics: These talks touched on advanced mathematical methods from dynamical systems, especially infinite dimensional ones arising with spatially heterogeneous problems (PDEs), asymptotic analysis, homogenization and averaging methods, numerical multiscale methods, methods from stochastic analysis and statistics. Apart from advancing disciplinary mathematical methods in these areas, in the present meeting also qualitatively new mathematical developments emerged: for example, mathematical and computational modelling of PDEs with stochastic data which are spatially inhomogeneous and do not satisfy stationarity or ergodic hypotheses. In processing experimental data (which becomes increasingly available at lower cost and, e.g. through modern scanning techniques, also at high volume and spatial and temporal resolution) new techniques of image and data processing have to be developed, and the mathematical models of complex systems have to allow for incorporation of statistical data extracted from these experiments. This has repercussions for the mathematical research and implies that novel algorithms are needed to generate computational grids adapted to voxel data. In the last five years mathematicians from analysis, stochastics and numerics started cooperation in this interdisciplinary field of research. New journals specifically devoted to these issues such as the SIAM Journal of Multiscale Analysis and Simulation, have been successfully launched. The previous meeting in Oberwolfach “Multiple Scale Systems – Modeling, Analysis and Numerics” from July 27 to August 2, 2003, gathered 42 scientists, among them approximately 15 junior scientists, from these areas. Since multiscale tools are crucial in many of the above themes, in the previous meeting mainly diffusion problems were treated. Reactive flow and transport, which were central themes in the present meeting, emerged only recently as key issues. The meeting was, exactly because of its wide scope, successful particularly in cross fertilizing different areas of applied mathematics and also raised a huge number of questions and challenges to participants documenting that the applications of mathematics to biological, social and other “complex systems” which has been emerging in the past years, is in the process of gaining momentum and, more importantly, stimulates development of new techniques and approaches in applied and computational mathematics at an increasing rate.

Abstract: Mathematical modeling of complex systems provides a rigorous framework for representing, analyzing, and predicting the behaviors that emerge from interacting components across diverse domains. This book develops an integrated conceptual and methodological framework that unites systems theory, nonlinear dynamics, stochastic processes, and computational intelligence to address complexity in biology, economics, and related fields. The problem addressed is the persistent gap between theoretical models and real-world decision-making, particularly under uncertainty, multi-scale interactions, and emergent phenomena. Methodologically, the work combines analytical modeling, simulation-based approaches, and hybrid theory–data integration, incorporating techniques such as epidemiological compartment models, biochemical reaction networks, agent-based economic simulations, and machine learning–augmented forecasting. The analysis demonstrates that cross-domain modeling not only uncovers structural parallels—such as feedback loops, network interdependencies, and adaptive behaviors—but also enables transferable tools for prediction, risk assessment, and policy design. Key results include a unified perspective on domain-specific methodologies, a framework for model validation and uncertainty quantification, and best-practice guidelines for computational enhancement and ethical deployment. The implications extend to improved predictive reliability in high-stakes contexts, from pandemic response and environmental management to macroeconomic stability and financial regulation. By bridging mathematical theory, computational innovation, and governance principles, this work positions modeling as both a scientific and strategic instrument for navigating complexity in the 21st century. Keywords: mathematical modeling, complex systems, nonlinear dynamics, stochastic processes, systems theory, computational modeling, biology, economics, epidemiological models, agent-based modeling, network theory, uncertainty quantification, sensitivity analysis, policy modeling, interdisciplinary frameworks, simulation, machine learning, emergent behavior, validation, ethical modeling

Background. Integration of reductionist and holistic approaches is one of the great challenges for mathematical modeling. Mathematical models of complex systems are divided into black-box, white-box and grey-box types. A black-box model is completely nonmechanistic as internal mechanisms of a modeled complex system are hidden. A white-box model demonstrates direct mechanisms of functioning of a complex system. It holistically shows all events at microlevel, mesolevel and macrolevel of a modeled system at all stages of its dynamics. Earlier we have used the white-box modeling for verification and reformulation of the competitive exlusion principle. Here we investigate our white-box model of single-species population dynamics. This is fundamentally important because most basic ecological models are of black-box type, including Malthusian, Verhulst, Lotka-Volterra models. Methods. Our white-box model of single-species population growth is a purely logical deterministic individual-based cellular automata model. A biological prototype of the model is a vegetative propagation of rhizomatous lawn grasses. Using the Monte Carlo method, we investigate a role of different initial positioning of an individual in the habitat. We also investigate different size and structure of the habitat and two types of fecundity. Results. We have created and investigated a logical white-box model of an ecosystem with one species. This model demonstrates mechanisms of the S-shaped and double S-shaped population growth. We have investigated population growth limited by different factors, in particular by resources, habitat structure, intraspecific competition, lifetime of individuals, regeneration time and fecundity of individuals. We have compared the S-shaped curves with J-shaped curves of population growth. Conclusion. We present a basic white-box model of population dynamics which combines reductionist and holistic approaches. Integration of reductionist and holistic approaches is provided by the simultaneous modeling of both part-whole and cause-effect relations in complex system. We consider this holystic multi-level white-box modeling approach as a method of artificial intelligence which works as hyper-logical automatic deductive inference that provides direct mechanistic insights into complex systems. The white-box modeling by logical deterministic cellular automata is a perspective way for investigation not only of population dynamics but also of any complex systems.

Common mathematical models of complex systems are not flexible, their creation is very resource-demanding and they are hard to work with. The numerous problems can arise during the process of building a mathematical model for complex systems. An area of knowledge, facts, and information could be structured badly or not structured at all. Part of the data might be missing or vice versa – we might have too much data available, which makes it difficult to find the necessary information. Therefore building a formal mathematical model, studying its dynamic for the relevant area of knowledge becomes a very hard or even almost impossible task. And that is why the new methods for such task are in much demand, namely, the methods of building descriptive mathematical models. The descriptive mathematical model serves as not a strict and formal model but a qualitative one. Such a qualitative model gives us a possibility to describe the character of the system, behavior of its internal components, and approximate rules of its dynamics. The qualitative model gives us a chance to deny the propositions, which do not fit the model directly at the first stage.

Coupling control and human factors in mathematical models of complex systems

Experimental identification as one of the possibilities of creating mathematical models of complex systems is a fast, simple and easy-to-use methodology. Even with a low percentage of knowledge of the modelled object (functionality, internal bonds, etc.) it is possible to obtain a mathematical model with high quality - similarity - verification. The disadvantage of experimental identification is the necessary existence of a functional complex system and a measuring - sensing apparatus. The Laboratory of intelligent control systems of aircraft engines has four full-featured functional systems - small and micro turbo compressor aviation jet engines. Experimental identification of small jet engines based on the measured transition characteristic is being researched using these engines. The question and topic of this article is to verify the possibilities and ways of creating mathematical models by experimental identification based on the acquired - measured transition characteristics for the smallest category of aircraft engines - micro turbo-compressor jet engine and evaluate obtained simulation models of different complexity.

Chapter 5 - Structured Mathematical Modeling

Recently, there has been substantial interest in the study of various random networks as mathematical models of complex systems. As these complex systems grow larger, the ability to generate progressively large random networks becomes all the more important. This motivates the need for efficient parallel algorithms for generating such networks. Naive parallelization of the sequential algorithms for generating random networks may not work due to the dependencies among the edges and the possibility of creating duplicate (parallel) edges. In this paper, we present MPI-based distributed memory parallel algorithms for generating random scale-free networks using the preferential-attachment model. Our algorithms scale very well to a large number of processors and provide almost linear speedups. The algorithms can generate scale-free networks with 50 billion edges in 123 seconds using 768 processors.

AFARS—an algorithm for analyzing the reliability of systems

The branch of complex system spans over a wide range of areas from physical and technological systems to social and biological systems. As the first step of any complex system analysis, modeling is an important task in scientific studies field. Then both the theory and practice of complex system modeling has been considered in recent years. It is well known that system identification is the theory and methods of establishing mathematical models of complex systems. Consequently, an identification approach for a class of complex nonlinear system is put forward in this paper. The idea of the identification method employs a system model composed with classical models so as to transform the system structure identification problem into a combinatorial optimization problem initially. Then, the artificial fish swarm optimization algorithm is used to synchronously implement the identification on the system’s structure and parameters. Finally, in simulation, compared with other algorithm, simulation results show that the proposed scheme is feasible.

Discrete-time stochastic systems with multiplicative and additive noises describe a wide class of mathematical models of complex systems, for example, industrial-technological, energy, economic, telecommunication, aerospace systems, etc. An important class of algorithms for processing measure ment information in complex systems are Kalman-type discrete-time filtering algorithms. Purpose. Construction of new discrete-time filtering algorithms for discrete-time linear systems with multiplicative and additive noises based on numerically stable modified weighted Gram – Schmidt orthogonalization (MWGS). Methodology. The methods of computational linear algebra were used, namely, the direct procedu re of MWGS-orthogonalization, the theory of Kalman filtering, methods of scientific programming in MATLAB. Findings. New LD-algorithms for discrete-time filtering in covariance and informational form for discrete-time stochastic systems with multiplicative and additive noises are constructed. The algorithms have an extended array form allowing updates for all necessary filter values. The method employs numerically stable modified weighted Gram – Schmidt orthogonalization. Algebraic equiva lence of LD-filters to Kalman-type covariance and informational algorithms for linear discrete time stochastic systems with multiplicative and additive noises is proved. The conducted numerical experiments have shown the effectiveness of the proposed algorithms using the example of solving the problem of parametric estimation of a model of almost rectilinear motion, as well as their savings in computation time compared to previously constructed UD filters. Value. New discrete-time filtering LD-algorithms can be used as a reliable computational alterna tive to the Kalman-type “standard algorithms” since they have the property being numerically stable to machine round-off errors due to the use of the MWGS orthogonalization computational procedure at each iteration of the algorithm. The results can be used to solve problems of measurement information processing in discrete-time systems with multiplicative and additive noise.

Complex systems may be mechanistically modelled by white-box modeling with using logical deterministic individual-based cellular automata. Mathematical models of complex systems are of three types: black-box (phenomenological), white-box (mechanistic, based on the first principles) and grey-box (mixtures of phenomenological and mechanistic models). Most basic ecological models are of black-box type, including Malthusian, Verhulst, Lotka–Volterra models. In black-box models, the individual-based (mechanistic) mechanisms of population dynamics remain hidden. Here we mechanistically model the S-shaped and double S-shaped population growth of vegetatively propagated rhizomatous lawn grasses. Using purely logical deterministic individual-based cellular automata we create a white-box model. From a general physical standpoint, the vegetative propagation of plants is an analogue of excitation propagation in excitable media. Using the Monte Carlo method, we investigate a role of different initial positioning of an individual in the habitat. We have investigated mechanisms of the single-species population growth limited by habitat size, intraspecific competition, regeneration time and fecundity of individuals in two types of boundary conditions and at two types of fecundity. Besides that, we have compared the S-shaped and J-shaped population growth. We consider this white-box modeling approach as a method of artificial intelligence which works as automatic hyper-logical inference from the first principles of the studied subject. This approach is perspective for direct mechanistic insights into nature of any complex systems.