# Consistent Dynamic Mode Decomposition

## TL;DR

This paper introduces a flexible, noise-robust variational approach for computing Dynamic Mode Decomposition matrices, applicable to nonlinear and small data scenarios, with efficient convergence and broad applicability.

## Contribution

A novel variational formulation for DMD that does not assume data structure, enabling analysis of nonlinear and small datasets with robustness to noise.

## Key findings

- Method outperforms existing techniques on benchmark systems.
- Approach is robust to noise and does not require sequential data.
- Converges empirically with efficient Sylvester equation solves.

## 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.