Abstract
Dynamic mode decomposition (DMD) is a popular data-driven framework to extract linear dynamics from complex high-dimensional systems. In this work, we study the system identification properties of DMD. We first show that DMD is invariant under linear transformations in the image of the data matrix. If, in addition, the data are constructed from a linear time-invariant system, then we prove that DMD can recover the original dynamics under mild conditions. If the linear dynamics are discretized with the Runge–Kutta method, then we further classify the error of the DMD approximation and detail that for one-stage Runge–Kutta methods; even the continuous dynamics can be recovered with DMD. A numerical example illustrates the theoretical findings.
Highlights
Dynamical systems play a fundamental role in many modern modeling approaches of physical and chemical phenomena
We take the view of a numerical analyst and assume that the data are obtained via time integration of the dynamics with a general Runge–Kutta method (RKM) with known order of convergence
In Theorem 3, we show that if the dynamic mode decomposition (DMD) approximation is constructed from data that are obtained via a RKM, the approximation error of DMD with respect to the ordinary differential equation is in the order of the error of the RKM
Summary
Dynamical systems play a fundamental role in many modern modeling approaches of physical and chemical phenomena. If a mathematical model is not available or not suited for modification, data-driven methods, such as the Loewner framework [6,7], vector fitting [8–10], operator inference [11], or dynamic mode decomposition (DMD) [12] may be used to create a low-dimensional realization directly from the measurement or simulation data of the system. Since DMD creates a discrete, linear time-invariant dynamical system from data, we are interested in answering the following questions: 2. Assume that the data used to generate the DMD approximation are obtained from a linear differential equation. It is essential to know how the data for the construction of the DMD model are generated to answer these questions. We take the view of a numerical analyst and assume that the data are obtained via time integration of the dynamics with a general Runge–Kutta method (RKM) with known order of convergence. The dashed lines represent the questions that we aim to answer in this paper
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
