Randomized Projection Learning Method forDynamic Mode Decomposition

TL;DR

This paper introduces a randomized projection approach to efficiently compute dynamic mode decomposition (DMD) in high-dimensional spaces, reducing computational costs while maintaining accuracy, based on the Johnson-Lindenstrauss Lemma.

Contribution

It generalizes DMD algorithms to incorporate random projections, enabling low-cost, high-quality mode estimation in reduced-dimensional spaces.

Findings

Random projections effectively estimate DMD modes.

The method reduces computational and storage costs.

Results align with theoretical guarantees of random projections.

Abstract

A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on projected space. In the spirit of Johnson-Lindenstrauss Lemma, we will use random projection to estimate the DMD modes in reduced dimensional space. In practical applications, snapshots are in high dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is infeasible, so our main computational goal is estimating the eigenvalue and eigenvectors of the DMD operator in a projected domain. We will generalize the current algorithm to estimate a projected DMD operator. We focus on a powerful and simple random projection algorithm that will reduce the computational and storage cost. While clearly, a random projection simplifies the algorithmic complexity of a detailed optimal projection, as we will show, generally the results can be excellent nonetheless, and quality understood through a well-developed theory of random projections. We will demonstrate that modes can be calculated for a low cost by the projected data with sufficient dimension. Keyword: Koopman Operator, Dynamic Mode Decomposition(DMD), Johnson-Lindenstrauss Lemma, Random Projection, Data-driven method.