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

Abstract

This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition. Searching this approximation in a data-driven approach is formalized as attempting to solve a low-rank constrained optimization 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 characterizes the $$\ell _2$$ -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 eigenvalue decomposition. The algorithms are evaluated by numerical simulations using synthetic and physical data benchmarks.