A kernel-based method for data-driven koopman spectral analysis

Abstract

A data-driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with highdimensional state spaces is presented. This approach approximates the Koopman operator using a set of scalar observables, which are functions that map states to scalars, that is determined implicitly by the choice of a kernel function. This circumvents the computational issues that arise due to the number of basis functions required to span a "sufficiently rich" subspace of the space of scalar observables in such applications. We illustrate this method on two examples: the FitzHugh-Nagumo PDE, a prototypical one-dimensional reaction-diffusion system, and vorticity data obtained from experimentally obtained velocity data for flow over a cylinder at Reynolds number 413. In both examples, we compare our results with related methods, such as Dynamic Mode Decomposition (DMD) that have the same cost as our approach.2010 Mathematics Subject Classification. Primary: 37M10, 65P99, 47B33.for PDEs, are intrinsic to a particular evolution law, and have temporal dynamics that are determined by their corresponding Koopman eigenvalues [5]; in effect, the Koopman modes look and act like the eigenvectors of a linear system even when the underlying evolution law is nonlinear.This representation is possible because the Koopman operator, which defines these quantities, is a linear operator and therefore can have eigenfunctions and eigenvalues, which are later used to define the modes. However, it acts on scalar observables, which are functions defined on state space, and is therefore an infinite-dimensional operator even for finite-dimensional dynamical systems. As a result, methods that approximate the Koopman operator (often implicitly) select a finite-dimensional subspace of the space of scalar observables to use during the computation. Currently, the most widely used method is Dynamic Mode Decomposition (DMD) [3][4][5][6]21] (along with its related modifications [8,10,20]), which implicitly select linear functions of the state as the basis functions [7]. This restrictive choice of basis allows DMD to be applied to large systems of ODEs or PDEs, but in many applications, this subspace is simply not "rich" enough to effectively approximate the Koopman operator [7,11].Other methods, such as Extended DMD [7], use an expanded set of observables, which can produce a more accurate approximation of the set of Koopman eigenvalues, eigenfunctions, and modes for nonlinear systems. For small systems of ODEs, a "sufficiently rich" subspace can often be spanned using a few hundred or thousand basis functions, which is computationally tractable even on a laptop. However, the size of the needed basis grows rapidly as the dimension of the state space increases, so the necessary computations quickly become infeasible. This explosion in the computational cost is common in machine learning applications, and one facet of the "curse of dimensionality" [15].In this manuscript, we introduce a data-driven, kernel-based met...

Data-driven approximation of the Koopman generator: Model reduction, system identification, and control

“…Data-driven approaches for the analysis of complex dynamical systems-be it methods to approximate transfer operators for computing metastable or coherent sets, methods to learn physical laws, or methods for optimization and control-have been steadily gaining popularity over the last years. Algorithms such as DMD [1,2], EDMD [3,4], SINDy [5], and their various kernel- [3,6,7], tensor- [8,9,10], or neural network-based [11,12,13] extensions and generalizations have been successfully applied to a plethora of different problems, including molecular and fluid dynamics, meteorology, finance, as well as mechanical and electrical engineering. An overview of different applications can be found, e.g., in [14].…”

On the Koopman Operator of Algorithms

“…The choice of the dictionary D is crucial for a successfuly approximation of the operator, see [21,42,43] and references therein. In this paper, we employ thin-plate radial basis functions [9],…”

Generalized Kernel-Based Dynamic Mode Decomposition

“…Tractable but Restrictive. To tackle the high-dimensional setting dim(H) p, authors propose to consider in their seminal work a specific class of mapping Ψ from R p to H [5]. They assume H to be a RKHS [12].…”

“…After discretisation of these equations, we obtain a discrete system with p = 4096 for the evolution of vorticity and temperature. The benchmark algorithms are: 1) low-rank DMD (LR-DMD) [13, Algorithm 3], 2) total-least-square DMD (TLS-DMD) [11], 3) kernel-based DMD (K-DMD) [5], 4) the proposed generalized kernel DMD (GK-DMD), i.e., Algorithm 1. For the K-DMD and GK-DMD algorithms, we use a quadratic polynomial kernel or a Gaussian kernel with a standard deviation of 10 [15].…”