Optimal Low-Rank Dynamic Mode Decomposition

TL;DR

This paper introduces a closed-form optimal solution for low-rank Dynamic Mode Decomposition, improving computational efficiency and accuracy in analyzing complex dynamical systems from experimental data.

Contribution

It proves the existence of a closed-form optimal solution and develops an SVD-based algorithm, surpassing sub-optimal methods in low-rank DMD.

Findings

The optimal solution can be computed explicitly using SVD.

The proposed algorithm outperforms existing sub-optimal methods.

Application to a toy example demonstrates improved performance.

Abstract

Dynamic Mode Decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of non-linear systems from experimental datasets. Recently, several attempts have extended DMD to the context of low-rank approximations. This extension is of particular interest for reduced-order modeling in various applicative domains, e.g. for climate prediction, to study molecular dynamics or micro-electromechanical devices. This low-rank extension takes the form of a non-convex optimization problem. To the best of our knowledge, only sub-optimal algorithms have been proposed in the literature to compute the solution of this problem. In this paper, we prove that there exists a closed-form optimal solution to this problem and design an effective algorithm to compute it based on Singular Value Decomposition (SVD). A toy-example illustrates the gain in performance of the proposed algorithm compared to state-of-the-art techniques.