A Modified Bisecting K-Means for Approximating Transfer Operators: Application to the Lorenz Equations

TL;DR

This paper explores the convergence of an extended dynamic mode decomposition method using a large nonlinear dictionary to analyze the Lorenz equations, focusing on operator approximation and system dynamics.

Contribution

It introduces a modified bisecting k-means algorithm to construct a scalable nonlinear dictionary for approximating transfer operators in complex dynamical systems.

Findings

The nonlinear dictionary effectively captures invariant measures and Koopman eigenfunctions.

Varying the dictionary size impacts the accuracy of autocorrelation representations.

The method scales efficiently to higher-dimensional systems.

Abstract

We investigate the convergence behavior of the extended dynamic mode decomposition for constructing a discretization of the continuity equation associated with the Lorenz equations using a nonlinear dictionary of over 1,000,000 terms. The primary objective is to analyze the resulting operator by varying the number of terms in the dictionary and the timescale. We examine what happens when the number of terms of the nonlinear dictionary is varied with respect to its ability to represent the invariant measure, Koopman eigenfunctions, and temporal autocorrelations. The dictionary comprises piecewise constant functions through a modified bisecting k-means algorithm and can efficiently scale to higher-dimensional systems.