Data-Driven Linearization of Dynamical Systems

TL;DR

This paper introduces a data-driven linearization method that improves upon DMD and EDMD by providing a more justified and higher-order local linearization of dynamical systems, especially within slow spectral submanifolds.

Contribution

It offers a new theoretical justification for DMD under generic conditions and develops a novel algorithm, DDL, for more accurate local linearization of complex dynamical systems.

Findings

DDL outperforms DMD and EDMD on numerical data

Theoretical justification for DMD under generic observables

Construction of linearizing transformations within spectral submanifolds

Abstract

Dynamic Mode Decomposition (DMD) and its variants, such as extended DMD (EDMD), are broadly used to fit simple linear models to dynamical systems known from observable data. As DMD methods work well in several situations but perform poorly in others, a clarification of the assumptions under which DMD is applicable is desirable. Upon closer inspection, existing interpretations of DMD methods based on the Koopman operator are not quite satisfactory: they justify DMD under assumptions that hold only with probability zero for generic observables. Here, we give a justification for DMD as a local, leading-order reduced model for the dominant system dynamics under conditions that hold with probability one for generic observables and non-degenerate observational data. We achieve this for autonomous and for periodically forced systems of finite or infinite dimensions by constructing linearizing transformations for their dominant dynamics within attracting slow spectral submanifolds (SSMs). Our arguments also lead to a new algorithm, data-driven linearization (DDL), which is a higher-order, systematic linearization of the observable dynamics within slow SSMs. We show by examples how DDL outperforms DMD and EDMD on numerical and experimental data.