Reprojection methods for Koopman-based modelling and prediction

TL;DR

This paper introduces a new geometric reprojection framework for Koopman-based models, improving prediction accuracy by respecting the underlying manifold structure in extended Dynamic Mode Decomposition.

Contribution

The paper proposes a novel geometric reprojector that enhances Koopman model accuracy while maintaining finite-data error bounds.

Findings

The geometric reprojector improves approximation accuracy.

It preserves the invariant manifold structure.

Finite-data error bounds are maintained.

Abstract

Extended Dynamic Mode Decomposition (eDMD) is a powerful tool to generate data-driven surrogate models for the prediction and control of nonlinear dynamical systems in the Koopman framework. In eDMD a compression of the lifted system dynamics on the space spanned by finitely many observables is computed, in which the original space is embedded as a low-dimensional manifold. While this manifold is invariant for the infinite-dimensional Koopman operator, this invariance is typically not preserved for its eDMD-based approximation. Hence, an additional (re-)projection step is often tacitly incorporated to improve the prediction capability. We propose a novel framework for consistent reprojectors respecting the underlying manifold structure. Further, we present a new geometric reprojector based on maximum-likelihood arguments, which significantly enhances the approximation accuracy and preserves known finite-data error bounds.