Data-driven Modeling of Dynamic Systems

Dynamic Mode Decomposition: Data-driven Modeling of Complex Systems—by J. Nathan Kutz, Steven L. Brunton, Bingni W. Brunton, and Joshua L. Proctor—was published by SIAM in 2016. It addresses the burgeoning field of data-driven dynamical systems and explores the dynamic mode decomposition (DMD).

DMD is a mathematical methodology that aims to distill interpretable and actionable insights from measurements of high-dimensional, complex systems. The mathematical, engineering, and scientific research communities have demonstrated the method’s broad applicability to applications in areas such as control theory, computer science, and fluid dynamics. They have also identified a fundamental theoretical connection to the analysis of nonlinear dynamical systems. Perhaps the most exciting recent development, however, is the translation of this method from academic research to the lexicon of standard techniques for data scientists and machine learning practitioners in industry settings. DMD has proven to be a surprisingly efficient and simple yet powerful computational method that fills an important technical gap in the analysis of high-dimensional measurement data from dynamically evolving systems. Our book describes the theoretical foundations of DMD and demonstrates the technique on a variety of application areas.

The following text comes from chapter one of Dynamic Mode Decomposition, entitled “Dynamic Mode Decomposition: An Introduction,” and has been modified slightly for clarity.

The data-driven modeling and control of complex systems is a rapidly evolving field with great potential to transform the engineering, biological, and physical sciences. There is unprecedented availability of high-fidelity measurements from historical records, numerical simulations, and experimental data; but while data is abundant, models often remain elusive. Modern systems of interest—such as turbulent fluids, epidemiological systems, networks of neurons, financial markets, or the climate—may be characterized as high-dimensional, nonlinear dynamical systems that exhibit rich multiscale phenomena in both space and time. However complex, many of these systems evolve on a low-dimensional attractor that one may characterize by spatiotemporal coherent structures. Here we introduce the topic of this book—dynamic mode decomposition (DMD)—which is a powerful new technique for the discovery of dynamical systems from high-dimensional data.

The DMD method originated in the fluid dynamics community as a method to decompose complex flows into a simple representation based on spatiotemporal coherent structures. Peter Schmid and Jörn Sesterhenn [6, 7] first defined the DMD algorithm and demonstrated its ability to provide insights from high-dimensional fluids data. The growing success of DMD stems from the fact that it is an equation-free, data-driven method capable of providing an accurate decomposition of a complex system into spatiotemporal coherent structures, which one may use for short-time future-state prediction and control. More broadly, DMD has quickly gained popularity since several studies [2-5] showed that it is connected to the underlying nonlinear dynamics through Koopman operator theory [1] and is readily interpretable with standard dynamical systems techniques.

The development of DMD is timely due to the concurrent rise of data science, which encompasses a broad range of techniques from machine learning and statistical regression to computer vision and compressed sensing. Improved algorithms, abundant data, vastly expanded computational resources, and interconnectedness of data streams make this a fertile ground for rapid development.

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References

[1] Koopman, B.O. (1931). Hamiltonian systems and transformation in Hilbert space. Proc. Nat. Acad. Sci., 17(5), 315-318.
[2] Mezić, I. (2005). Spectral properties of dynamical systems, model reduction and decompositions. Nonlin. Dynam., 41(1-3), 309-325. 
[3] Mezić, I. (2013). Analysis of fluid flows via spectral properties of the Koopman operator. Ann. Rev. Fluid Mech., 45, 357-378. 
[4] Mezić, I., & Banaszuk, A. (2004). Comparison of systems with complex behavior. Physica D: Nonlin. Phenom., 197(1-2), 101-133.
[5] Rowley, C.W., Mezić, I., Bagheri, S., Schlatter, P., & Henningson, D.S. (2009). Spectral analysis of nonlinear flows. J. Fluid Mech., 645, 115-127.
[6] Schmid, P. (2010). Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech., 656, 5-28.
[7] Schmid, P.J., & Sesterhenn, J. (2008). Dynamic mode decomposition of numerical and experimental data. In 61st annual meeting of the APS division of fluid dynamics. San Antonio, TX: American Physical Society.