Low-Rank Dynamic Mode Decomposition: An Exact and Tractable Solution

TL;DR

This paper introduces a closed-form, polynomial-time solution for low-rank dynamic mode decomposition, enabling efficient and optimal linear approximations of high-dimensional dynamical systems with practical algorithms.

Contribution

It provides the first exact, tractable solution to the low-rank DMD optimization problem, improving over existing sub-optimal methods.

Findings

Closed-form solution for low-rank DMD obtained

Algorithms built from the optimal solution demonstrate efficiency

Numerical simulations validate the approach on synthetic and physical data

Abstract

This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach is formalised as attempting to solve a low-rank constrained optimisation problem. This problem is non-convex and state-of-the-art algorithms are all sub-optimal. This paper shows that there exists a closed-form solution, which is computed in polynomial time, and characterises the l2-norm of the optimal approximation error. The paper also proposes low-complexity algorithms building reduced models from this optimal solution, based on singular value decomposition or eigen value decomposition. The algorithms are evaluated by numerical simulations using synthetic and physical data benchmarks.