Extraction and prediction of coherent patterns in incompressible flows through space–time Koopman analysis

Abstract

We develop methods for detecting and predicting the evolution of coherent spatiotemporal patterns in incompressible time-dependent fluid flows driven by ergodic dynamical systems. Our approach is based on representations of the generators of the Koopman and Perron-Frobenius groups of operators governing the evolution of observables and probability measures on Lagrangian tracers, respectively, in a smooth orthonormal basis learned from velocity field snapshots through the diffusion maps algorithm. These operators are defined on the product space between the state space of the fluid flow and the spatial domain in which the flow takes place, and as a result their eigenfunctions correspond to global space-time coherent patterns under a skew-product dynamical system. Moreover, using this data-driven representation of the generators in conjunction with Leja interpolation for matrix exponentiation, we construct model-free prediction schemes for the evolution of observables and probability densities defined on the tracers. We present applications to periodic Gaussian vortex flows and aperiodic flows generated by Lorenz 96 systems.

Delay-Coordinate Maps and the Spectra of Koopman Operators

“…Also of interest (and in some cases easier to compute) are the projections of the observation map F onto the Koopman eigenfunctions, called Koopman modes [50]. Data-driven techniques for computing Koopman eigenvalues, eigenfunctions, and modes that have been explored in the past include methods based on generalized Laplace analysis [50,52], dynamic mode decomposition (DMD) [55,57,58,62], extended DMD (EDMD) [40,67], Hankel matrix analysis [2,14,62], and data-driven Galerkin methods [30,33,34]. The latter approach, as well as the related work in [8], additionally address the problem of nonparametric prediction of observables and probability densities.…”

“…At the same time, spaces of observables are also infinite dimensional, so the issue of finitedimensional approximation of (potentially unbounded) operators becomes relevant.Starting from the techniques proposed in [22,50,52], the operator-theoretic approach to ergodic theory has stimulated the development of a broad range of techniques for data-driven modeling of dynamical systems. These methods employ either the Koopman [2,14,15,30,33,34,40,41,50,52,55,57,62,67] or the Perron-Frobenius (transfer) operators [21,22,[27][28][29], which are duals to one another in appropriate function spaces. The goal common to these techniques is to approximate spectral quantities for the operator in question, such as eigenvalues, eigenfunctions, and spectral projections, from measured values of observables along orbits of the dynamics.…”

“…The goal common to these techniques is to approximate spectral quantities for the operator in question, such as eigenvalues, eigenfunctions, and spectral projections, from measured values of observables along orbits of the dynamics. To that end, a diverse range of approaches has been employed, including state space partitions [21,22,[27][28][29], harmonic averaging [20,50,52], iterative methods [55,57], dictionary/basis representations [30,33,34,40,62,67], delay-coordinate embeddings [2,14,30,33], and spectral-moment estimation [41].Compared to observables identified by eigendecomposition techniques based on kernel integral operators that do not depend on the dynamics (e.g., covariance [3,38] or heat operators [5,11,16], the latter of which have been popular in manifold learning applications), eigenfunctions of evolution operators are likely to offer higher physical interpretability and predictability, as they are determined from an operator intrinsic to the dynamical system. In particular, one of the key properties of Koopman or Perron-Frobenius eigenfunctions for ergodic dynamical systems is that they evolve periodically and with a single frequency (even if the underlying dynamical system is aperiodic), and thus have high predictability.…”

Data-driven Koopman operator approach for computational neuroscience

“…We apply a recently developed framework [15,23,24,32] inspired from the operatortheoretic formulation of ergodic theory that addresses many of the limitations of PCA and related approaches outlined above. Instead of estimating a covariance operator, our approach is based on data-driven approximations of the Koopman operator: the fundamental operator governing the evolution of observables (functions of the state) of a dynamical system.…”

“…These oscillations are commonly divided into 5-8 bands based on their frequency. The oscillations vary from slow, e.g., Delta (1)(2)(3)(4) and Theta (4)(5)(6)(7)(8), to medium like Alpha (8)(9)(10)(11)(12), to quickly oscillating modes such as Beta (12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22)(23)(24)(25)(26)(27)(28)(29)(30) and Gamma . Electrophysiologic recordings, which allow researchers to capture distinct oscillations, are one of the most affordable and readily-accessible sources of data on brain activation patterns.…”