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Data based identification and prediction of nonlinear and complex dynamical systems. (English) Zbl 1359.70089
Summary: The problem of reconstructing nonlinear and complex dynamical systems from measured data or time series is central to many scientific disciplines including physical, biological, computer, and social sciences, as well as engineering and economics. The classic approach to phase-space reconstruction through the methodology of delay-coordinate embedding has been practiced for more than three decades, but the paradigm is effective mostly for low-dimensional dynamical systems. Often, the methodology yields only a topological correspondence of the original system. There are situations in various fields of science and engineering where the systems of interest are complex and high dimensional with many interacting components. A complex system typically exhibits a rich variety of collective dynamics, and it is of great interest to be able to detect, classify, understand, predict, and control the dynamics using data that are becoming increasingly accessible due to the advances of modern information technology. To accomplish these goals, especially prediction and control, an accurate reconstruction of the original system is required.
Nonlinear and complex systems identification aims at inferring, from data, the mathematical equations that govern the dynamical evolution and the complex interaction patterns, or topology, among the various components of the system. With successful reconstruction of the system equations and the connecting topology, it may be possible to address challenging and significant problems such as identification of causal relations among the interacting components and detection of hidden nodes. The “inverse” problem thus presents a grand challenge, requiring new paradigms beyond the traditional delay-coordinate embedding methodology.
The past fifteen years have witnessed rapid development of contemporary complex graph theory with broad applications in interdisciplinary science and engineering. The combination of graph, information, and nonlinear dynamical systems theories with tools from statistical physics, optimization, engineering control, applied mathematics, and scientific computing enables the development of a number of paradigms to address the problem of nonlinear and complex systems reconstruction. In this Review, we describe the recent advances in this forefront and rapidly evolving field, with a focus on compressive sensing based methods. In particular, compressive sensing is a paradigm developed in recent years in applied mathematics, electrical engineering, and nonlinear physics to reconstruct sparse signals using only limited data. It has broad applications ranging from image compression/reconstruction to the analysis of large-scale sensor networks, and it has become a powerful technique to obtain high-fidelity signals for applications where sufficient observations are not available. We will describe in detail how compressive sensing can be exploited to address a diverse array of problems in data based reconstruction of nonlinear and complex networked systems. The problems include identification of chaotic systems and prediction of catastrophic bifurcations, forecasting future attractors of time-varying nonlinear systems, reconstruction of complex networks with oscillatory and evolutionary game dynamics, detection of hidden nodes, identification of chaotic elements in neuronal networks, reconstruction of complex geospatial networks and nodal positioning, and reconstruction of complex spreading networks with binary data.. A number of alternative methods, such as those based on system response to external driving, synchronization, and noise-induced dynamical correlation, will also be discussed.
Due to the high relevance of network reconstruction to biological sciences, a special section is devoted to a brief survey of the current methods to infer biological networks. Finally, a number of open problems including control and controllability of complex nonlinear dynamical networks are discussed. The methods outlined in this Review are principled on various concepts in complexity science and engineering such as phase transitions, bifurcations, stabilities, and robustness. The methodologies have the potential to significantly improve our ability to understand a variety of complex dynamical systems ranging from gene regulatory systems to social networks toward the ultimate goal of controlling such systems.

MSC:
70K55 Transition to stochasticity (chaotic behavior) for nonlinear problems in mechanics
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
37F99 Dynamical systems over complex numbers
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