A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

TL;DR

This paper introduces a data-driven extension of Dynamic Mode Decomposition for approximating Koopman operator eigenvalues, eigenfunctions, and modes from snapshot data, applicable to both deterministic and stochastic systems.

Contribution

It presents a novel method that extends DMD to approximate Koopman spectral properties without requiring explicit models or black-box simulations.

Findings

Effective approximation of Koopman eigenvalues and modes from data.

Handles both deterministic and stochastic data, approximating stochastic Koopman eigenfunctions.

Demonstrated through four illustrative examples showing practical applications.

Abstract

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.