Abstract. We propose a new method for computing Dynamic Mode Decomposition (DMD) evolution matrices, which we use to analyze dynamical systems. Unlike the majority of existing methods, our approach is based on a variational formulation consisting of data alignment penalty terms and constitutive orthogonality constraints. Our method does not make any assumptions on the structure of the data or their size, and thus it is applicable to a wide range of problems including non-linear scenarios or extremely small observation sets. In addition, our technique is robust to noise that is independent of the dynamics and it does not require input data to be sequential. Our key idea is to introduce a regularization term for the forward and backward dynamics. The obtained minimization problem is solved efficiently using the Alternating Method of Multipliers (ADMM) which requires two Sylvester equation solves per iteration. Our numerical scheme converges empirically and is similar to a provably convergent ADMM scheme. We compare our approach to various state-of-the-art methods on several benchmark dynamical systems. Key words. Dynamic Mode Decomposition, Dynamical Systems, ADMM, variational formulation AMS subject classifications. 37N30, 65K10, 90C26 1. Introduction. Over the last few years, data-driven approaches became prevalent in analyzing dynamical systems [22] . In the common scenario, a collection of system observations is provided and a linear object that encodes the dynamics is generated based solely on the data. These data-driven approaches are advantageous in that they make minimal assumptions on the governing equations of the system, and in particular, these techniques are applicable even to non-linear dynamics. In this context, Dynamic Mode Decomposition (DMD) [30] methods gained a lot of attention lately, in part due to their computational efficiency as well as their analysis capabilities of the system at hand. DMD-based methods were successfully applied to various flows including detonation waves, cavity flows and jets [25, 32, 31] . In short, DMD computes a matrix whose spectrum, represented by the eigenvalues and eigenvectors, provides meaningful information such as growth and decay rates of the system or dominant coherent structures in the flow. The goal of this paper is to propose a new method for computing DMD matrices that is based on interpreting the problem in a variational form, taking into account the forward and backward dynamics and solving it efficiently via splitting. Developing data-driven methodologies for the analysis of non-linear dynamical systems is an active research domain with DMD being one of its main avenues. In particular, DMD was recently generalized and extended in several works having the objective of alleviating some of the shortcomings in the original technique. For instance, a limiting assumption
While many extensions of the original algorithm have been proposed, most related to our approach is the work of REF where the authors consider the forward and backward dynamics in a non-neural optimization setting.
Consistent Dynamic Mode Decomposition
SIAM Journal on Applied Dynamical Systems
SIAM Journal on Applied Dynamical Systems
Mathematics Computer Science