The Koopman operator is a linear but infinite dimensional operator that governs the evolution of scalar observables defined on the state space of an autonomous dynamical system, and is a powerful tool for the analysis and decomposition of nonlinear dynamical systems. In this manuscript, we present a data driven method for approximating the leading eigenvalues, eigenfunctions, and modes of the Koopman operator. The method requires a data set of snapshot pairs and a dictionary of scalar observables, but does not require explicit governing equations or interaction with a "black box" integrator. We will show that this approach is, in effect, an extension of Dynamic Mode Decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes. Furthermore, if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation, which could be considered as the "stochastic Koopman operator" [1]. Finally, four illustrative examples are presented: two that highlight the quantitative performance of the method when presented with either deterministic or stochastic data, and two that show potential applications of the Koopman eigenfunctions.

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...

Dynamic mode decomposition (DMD) provides a practical means of extracting insightful dynamical information from fluids datasets. Like any data processing technique, DMD's usefulness is limited by its ability to extract real and accurate dynamical features from noise-corrupted data. Here we show analytically that DMD is biased to sensor noise, and quantify how this bias depends on the size and noise level of the data. We present three modifications to DMD that can be used to remove this bias: (i) a direct correction of the identified bias using known noise properties, (ii) combining the results of performing DMD forwards and backwards in time, and (iii) a total least-squares-inspired algorithm. We discuss the relative merits of each algorithm, and demonstrate the performance of these modifications on a range of synthetic, numerical, and experimental datasets. We further compare our modified DMD algorithms with other variants proposed in recent literature.

We formulate a low-storage method for performing dynamic mode decomposition that can be updated inexpensively as new data become available; this formulation allows dynamical information to be extracted from large datasets and data streams. We present two algorithms: the first is mathematically equivalent to a standard "batch-processed" formulation; the second introduces a compression step that maintains computational efficiency, while enhancing the ability to isolate pertinent dynamical information from noisy measurements. Both algorithms reliably capture dominant fluid dynamic behaviors, as demonstrated on cylinder wake data collected from both direct numerical simulations and particle image velocimetry experiments.Dynamic mode decomposition (DMD) is a data-driven computational technique capable of extracting dynamical information from flowfields measured in physical experiments or generated by direct numerical simulations. 1 Since its introduction in 2008, 2 DMD has been used in the analysis of numerous fluid mechanical systems (e.g., bluff body flows, 3 jet flows, 4,5 and viscoelastic fluid flows 6 ) and has gained increasing popularity owing to its ability to reveal and quantify the dynamics of a flow, even when those dynamics are nonlinear. 4,7 DMD operates on snapshots of the flowfield (e.g., velocity, vorticity, pressure) and their time-shifted counterparts-obtained either from experiments or numerical simulations-to compute the eigenvalues ("DMD eigenvalues") and eigenvectors ("DMD modes") of a linear operator that best fits the associated dynamics in a least-squares sense. The DMD modes represent spatial fields that often highlight coherent structures in the flow, while the associated DMD eigenvalues dictate the decay/growth rates and oscillation frequencies of these modes. As such, access to DMD modes and eigenvalues enables a reconstruction of the dynamics associated with a given flowfield. Other modal decomposition techniques, such as the commonly employed proper orthogonal decomposition (POD), only compute spatial modes associated with the flow. 8 Although spatial modes can offer valuable information regarding coherent structures and other flow qualities (e.g., in the case of POD, they determine the most energetic modes), characterizing the underlying dynamics relies upon projecting these spatial modes onto an assumed dynamical form. DMD offers an advantage over these other modal decomposition techniques in that it computes both spatial modes and their associated temporal behaviors, thus removing any guesswork associated with realizing a dynamical representation of the system.To date, researchers have viewed DMD as a post-processing tool; that is, a method that requires the entire experimental or computational dataset to be available prior to commencing

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