Analytic Extended Dynamic Mode Decomposition

TL;DR

This paper introduces analytic EDMD, a novel algorithm that accurately captures the Koopman spectrum for analytic functions using a data-driven Taylor approximation, with proven convergence and error bounds.

Contribution

It develops a new EDMD variant leveraging polynomial projections and Taylor approximations, improving spectral accuracy and avoiding pollution in Koopman analysis.

Findings

Accurately captures Koopman spectrum for hyperbolic equilibria

Provides convergence proofs and error bounds for eigenfunctions and spectrum

Demonstrates superior performance through numerical examples

Abstract

We develop a novel EDMD-type algorithm that captures the spectrum of the Koopman operator defined on a reproducing kernel Hilbert space of analytic functions. This method, which we call analytic EDMD, relies on an orthogonal projection on polynomial subspaces, which is equivalent to a data-driven Taylor approximation. In the case of dynamics with a hyperbolic equilibrium, analytic EDMD demonstrates excellent performance to capture the lattice-structured Koopman spectrum based on the eigenvalues of the linearized system at the equilibrium. Moreover, it yields the Taylor approximation of associated principal eigenfunctions. Since the method preserves the triangular structure of the operator, it does not suffer from spectral pollution and, moreover, arbitrary accuracy on the spectrum can be reached with a fixed finite dimension of the approximation and with a (possibly non-uniform) sampling over an arbitrary set of nonzero measure. The performance of analytic EDMD is illustrated with numerical examples and is assessed through a comparative study with related methods. Finally, the method is complemented with theoretical results, proving strong convergence of the eigenfunctions and providing error bounds on the spectrum estimation.