PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operator

Engineers have long relied on linearization to bridge the gap between simplified, linear descriptions where powerful analytical tools exist, and the intricate complexities of nonlinear dynamics where analytical solutions are elusive [5, 6]. Local linearization, implemented via first-order Taylor series approximation, has been widely used in system identification [5], optimization [6], and many other fields to make problems tractable. However, many real-world systems are fundamentally nonlinear and require solutions outside of the local neighborhood where linearization is valid. Rapid progress in machine learning and big data methods are driving advances in the data-driven modeling of such nonlinear systems in science and engineering [7] Koopman operator theory in particular has emerged as a principled approach to embed nonlinear dynamics in a linear framework that goes beyond simple linearization [4]. In the diverse landscape of data-driven modeling approaches, Koopman operator theory has received considerable attention in recent years [8-13]. These strategies encompass not only linear methodologies [5, 14] and dynamic mode decomposition (DMD) [1, 2, 15], but also more advanced techniques such as nonlinear autoregressive algorithms [16, 17], neural networks [18-27], Gaussian process regression [28], operator inference, and reduced-order modeling [29-31], among others [32-38]. The Koopman operator perspective is unique within data-driven modeling techniques due to its distinct aim of learning a coordinate system in which the nonlinear dynamics become linear. This methodology enables the application of closed-form, convergence-guaranteed methods from linear system theory to general nonlinear dynamics. To fully leverage the potential of data-driven Koopman theory across a diverse range of scientific and engineering disciplines, it is critical to have a central toolkit to automate state-of-the-art Koopman operator algorithms.