Time-Delay Observables for Koopman: Theory and Applications

Abstract

Nonlinear dynamical systems are ubiquitous in science and engineering, yet many issues still exist related to the analysis and prediction of these systems. Koopman theory circumvents these issues by transforming the finite-dimensional nonlinear dynamics to a linear dynamical system of functions in an infinite-dimensional Hilbert space of observables. The eigenfunctions of the Koopman operator evolve linearly in time and thus provide a natural coordinate system for simplifying the dynamical behaviors of the system. We consider a family of observable functions constructed by projecting the delay coordinates of the system onto the eigenvectors of the autocorrelation function, which can be regarded as continuous SVD basis vectors for time-delay observables. We observe that these functions are the most parsimonious basis of observables for a system with Koopman mode decomposition of order N , in the sense that the associated Koopman eigenfunctions are guaranteed to lie in the span of the first N of these coordinates. We conjecture and prove a number of theoretical results related to the quality of these approximations in the more general setting where the system has mixed spectra or the coordinates are otherwise insufficient to capture the full spectral information. We prove a related and very general result that the dynamics of the observables generated by projecting delay coordinates onto an arbitrary orthonormal basis are systemindependent and depend only on the choice of basis, which gives a highly efficient way of computing representations of the Koopman operator in these coordinates. We show that this formalism provides a theoretical underpinning for the empirical results in [8], which found that chaotic dynamical systems can be approximately factored into intermittently forced linear systems when viewed in delay coordinates. Finally, we compute these time delay observables for a number of example dynamical systems and show that empirical results match our theory.

7 Data-driven methods for reduced-order modeling

“…Instead of advancing instantaneous linear or nonlinear measurements of the state of a system directly, as in DMD, it may be possible to obtain intrinsic measurement coordinates for Koopman based on time-delayed measurements of the system [127,22,5,43,68]. This perspective is data-driven, relying on the wealth of information from previous measurements to inform the future.…”

“…One version of time-delay embeddings, the HAVOK, has been used successfully to diagnose a diverse set of dynamical systems [22]. More broadly, there are a number of analysis tools that can be applied to the Hankel matrix for analysis of dynamics [68]. The time-delay measurement scheme is shown schematically in Figure 7.2, as illustrated on the Lorenz system for a single time-series measurement of the first variable, x(t).…”

Sparsity-promoting algorithms for the discovery of informative Koopman-invariant subspaces

“…Time-delay embedding can be very useful in reduced-order modelling of systems for which sparse measurements can be easily obtained, assuming the inputs and outputs are not high dimensional (Korda & Mezić 2018a). Although time-delay embedding is simple to implement and has strong connections to Takens' embedding (Kamb et al 2018;Pan & Duraisamy 2019), the main practical issue arises in reduced-order modelling of high-fidelity simulations in a predictive setting due to the requirement of a large number of snapshots of the full-order model. Furthermore, if one is only interested in the post-transient dynamics of the system state on an attractor, linear observables with time delays are sufficient to extract an informative Koopman-invariant subspace (Mezić 2005;Arbabi & Mezić 2017a,b;Brunton et al 2017;Röjsel 2017;Pan & Duraisamy 2019).…”

Revealing the Dynamical Transition of Anisotropy Behind the HOST by Koopman Analysis

“…We focus on a cutting‐edge data‐driven method, an advanced approach to Koopman analysis based on the ergodic theory about the dynamical system using the time delay embedding technique, which is called the Hankel alternative view of Koopman analysis (HAVOK) (Arbabi & Mezić, 2017b; Brunton et al, 2016). Established on the invariant subspaces diffeomorphisms with the original dynamical system (Kamb et al, 2020; Takens, 1981), this method can estimate the complex dynamical system from a single point measurement time series (Champion et al, 2019), which can be adopted to investigate the single tower observation data.…”