Modeling of dynamical systems through deep learning
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REVIEW
Modeling of dynamical systems through deep learning P. Rajendra 1
&
V. Brahmajirao 2
Received: 15 July 2020 / Accepted: 8 November 2020 # International Union for Pure and Applied Biophysics (IUPAB) and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract This review presents a modern perspective on dynamical systems in the context of current goals and open challenges. In particular, our review focuses on the key challenges of discovering dynamics from data and finding data-driven representations that make nonlinear systems amenable to linear analysis. We explore various challenges in modern dynamical systems, along with emerging techniques in data science and machine learning to tackle them. The two chief challenges are (1) nonlinear dynamics and (2) unknown or partially known dynamics. Machine learning is providing new and powerful techniques for both challenges. Dimensionality reduction methods are used for projecting dynamical methods in reduced form, and these methods perform computational efficiency on real-world data. Data-driven models drive to discover the governing equations and give laws of physics. The identification of dynamical systems through deep learning techniques succeeds in inferring physical systems. Machine learning provides advanced new and powerful algorithms for nonlinear dynamics. Advanced deep learning methods like autoencoders, recurrent neural networks, convolutional neural networks, and reinforcement learning are used in modeling of dynamical systems. Keywords Dynamical systems . Machine learning . Dimensionality reduction . Deep learning . Dynamic mode decomposition
Introduction Dynamical systems provide a mathematical framework to describe the real-world problems, modeling the rich interactions among quantities that change in time. Formally, dynamical systems concern the analysis, prediction, and understanding of the behavior of systems of differential equations or iterative mappings that describe the evolution of the state of a system. Data-driven models are an emerging field of simulating and discovering dynamical systems purely from data using techniques of machine learning and data science. We have an explosion of data in climate science, neuroscience, disease modeling, and fluid dynamics. The amount of data getting from experiments, simulations, and historical records is growing at an incredible pace. Simultaneously, the algorithms in
* P. Rajendra [email protected] 1
Department of Mathematics CMR Institute of Technology, Bengaluru, India
2
School of Biotechnology MGNIRSA, D.S.R. Foundation, Hyderabad, India
machine learning, data science, and statistics optimization techniques are getting much better. Therefore we can discover dynamical systems and characterize them purely from data. In the past, dynamical systems were essentially written down by physical laws and derived the equations from first principles using physics. But, today the systems that we want to understand like the brain, climate, or financial market. There are no first
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