[Paper Review] Optimal Low-Rank Dynamic Mode Decomposition

Patrick Héas, Cédric Herzet|arXiv (Cornell University)|Jan 4, 2017

Model Reduction and Neural Networks17 references12 citations

TL;DR

This paper presents a closed-form optimal solution for low-rank Dynamic Mode Decomposition (DMD) via Singular Value Decomposition (SVD), overcoming the sub-optimality of prior iterative or assumption-laden methods. It enables exact computation of the rank-k DMD approximation without requiring linear dependence assumptions, significantly improving accuracy over state-of-the-art techniques, especially in non-linear or non-ideal settings.

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.

Motivation & Objective

• To address the lack of optimal solutions for the non-convex low-rank DMD approximation problem.
• To eliminate reliance on iterative solvers or restrictive assumptions like linear dependence of snapshots.
• To develop a computationally efficient, analytically optimal algorithm for reduced-order modeling.
• To provide a framework that enables accurate, closed-form computation of DMD modes and amplitudes.

Proposed method

• Formulates the low-rank DMD problem as a non-convex optimization: minimize ‖Y − AX‖_F subject to rank(A) ≤ k.
• Derives a closed-form solution using the Eckart-Young theorem via SVD of the data matrices X and Y.
• Constructs the optimal low-rank matrix Â_k using the SVD of X and the projection matrix P from the SVD of Y.
• Employs a two-stage algorithm: first compute the optimal A⋆_k via SVD and matrix projections, then extract DMD modes and amplitudes via SVD of a reduced matrix.
• Avoids iterative refinement or assumptions about linear dependence between snapshots.
• Enables efficient reduced-order modeling by computing DMD modes and amplitudes directly from the SVD of a low-dimensional matrix.

Experimental results

Research questions

• RQ1 Can a closed-form optimal solution be derived for the non-convex low-rank DMD approximation problem?
• RQ2 Does the proposed method outperform existing iterative or assumption-based approaches in accuracy and robustness?
• RQ3 Can the solution be computed efficiently without iterative optimization or restrictive linear dependence assumptions?
• RQ4 How does the performance of the optimal solution compare to truncated full-rank DMD or projected DMD in non-linear systems?
• RQ5 Is the proposed method robust across different system dynamics, including linear and non-linear cases?

Key findings

• The proposed algorithm achieves the lowest error norm ∥Y − Â_kX∥_F across all tested settings, confirming its optimality.
• In non-linear systems (setting iii), the proposed method (a) significantly outperforms projected DMD (c), with error increasing by over 1000% for k > 10.
• For linear systems satisfying the linear dependence assumption (setting i), the proposed method matches the performance of projected DMD (c), confirming its optimality in ideal cases.
• The truncated full-rank DMD solution (b) performs poorly when k < 30, indicating sub-optimality of simple truncation.
• All methods correctly identify the underlying low-dimensional subspace when k ≥ r = 30, confirming the method’s consistency with the true system rank.
• The algorithm achieves optimal performance without iterative refinement or restrictive modeling assumptions, proving the existence of a closed-form solution.