Deep Learning Enhanced Dynamic Mode Decomposition

TL;DR

This paper introduces DLDMD, a novel method combining deep learning and dynamic mode decomposition to improve the analysis and prediction of nonlinear dynamical systems by learning optimal observables.

Contribution

The paper proposes DLDMD, integrating autoencoders with EDMD to automatically discover optimal observables for better flow embedding and prediction in nonlinear systems.

Findings

DLDMD outperforms standard DMD in nonlinear data prediction.

The method enables accurate flow embedding and state prediction.

It automates observable selection, reducing user intervention.

Abstract

Koopman operator theory shows how nonlinear dynamical systems can be represented as an infinite-dimensional, linear operator acting on a Hilbert space of observables of the system. However, determining the relevant modes and eigenvalues of this infinite-dimensional operator can be difficult. The extended dynamic mode decomposition (EDMD) is one such method for generating approximations to Koopman spectra and modes, but the EDMD method faces its own set of challenges due to the need of user defined observables. To address this issue, we explore the use of autoencoder networks to simultaneously find optimal families of observables which also generate both accurate embeddings of the flow into a space of observables and submersions of the observables back into flow coordinates. This network results in a global transformation of the flow and affords future state prediction via the EDMD and the decoder network. We call this method the deep learning dynamic mode decomposition (DLDMD). The method is tested on canonical nonlinear data sets and is shown to produce results that outperform a standard DMD approach and enable data-driven prediction where the standard DMD fails.