Koopman eigenfunction approximations by a least-squares Galerkin method

In recent years, much attention and research has been devoted to the Koopman operator following the seminal 2005 work [1] by Mesic. The Koopman operator offers the possibility to exactly transform any dynamical system described by a (sufficiently smooth) autonomous nonlinear differential equation for its states into a (generally infinite-dimensional) decoupled linear system. More precisely, the Koopman operator is a (semi)group of linear operators with a generator that describes the so-called Kolmogorov forward equation. The time dependence of any eigenfunction φ of the Koopman operator (and its generator) is given by dφ/dt = λφ, where λ is the corresponding eigenvalue.
Much of the current research focuses on data-driven methods for system modeling and identification, spurred by the development of dynamic mode decomposition [2] and its extensions [3] in the fluid dynamics community. These methods are algorithms for estimating eigenvalues and eigenfunctions of the Koopman operator from time series data. The core idea is a least-squares regression of ansatz functions (often referred to as a library) to map many initial states x(0) to their corresponding states x(T) after the same time interval T, resulting from the flow of the dynamical system. ... mehrOften the time series data come from simulations, not measurements, and these data-driven methods are used as sophisticated post-processing methods to identify time-periodic patterns. Since the choice of appropriate ansatz functions is non-trivial, a number of extensions to the use of neural networks as universal function approximators have been investigated in recent years.
Given a system model in the form of a state-space equation and a set of ansatz functions, approximations of the Koopman eigenfunctions can also be computed using a classical Galerkin method. Since the focus of most research in Koopman operators is data-driven, this approach has not been studied as extensively. In order to compare a Galerkin approximation in terms of performance and computational effort with established data-driven methods such as (extended) dynamic mode decomposition and neural networks (in particular as autoencoders), we analyze the Koopman operator of a damped pendulum – a well-studied mechanical system whose simplicity permits interpretation of the results.

[1] Mezić, I.: Spectral properties of dynamical systems, model reduction and decompositions. Nonlinear Dynamics 41(1–3), 309–325 (2005)
[2] Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. Journal of Fluid Mechanics 65(6), 5–28 (2010)
[3] Tu, J.H. et al.: On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics 1(2), 391–421 (2014)