Weak-DMD: A Galerkin approach to the problem of noise in the Dynamic Mode Decomposition algorithm

TL;DR

Weak-DMD introduces a Galerkin-based weak formulation of Dynamic Mode Decomposition that mitigates noise effects and relaxes timestep constraints, demonstrated on nuclear engineering and fluid flow problems.

Contribution

The paper proposes weak-DMD, a novel Galerkin approach to improve DMD's robustness to noise and sampling issues, extending its applicability.

Findings

Weak-DMD effectively filters measurement noise in DMD.

Weak-DMD performs well on nuclear engineering and fluid flow datasets.

Compared to standard DMD, weak-DMD shows improved accuracy in noisy conditions.

Abstract

Dynamic Mode Decomposition (DMD) is a data-driven method for approximating the spatiotemporal modes of a system. The eigenvectors and eigenvalues of the system are approximated from a series of time-snapshots of the state variables. The standard formulation of DMD is subject to strict assumptions concerning the time-spacing of the snapshots and is biased by measurement noise. Variations on the method have been developed to address these shortcomings, but the problem is still open. Motivated by the effectiveness of Galerkin methods in the field of model discovery, a weak formulation of DMD is presented, weak-DMD. Weak-DMD precludes timestep considerations and also filters noise. Results for two nuclear engineering applications and the flow of fluid past a cylinder are given and compared with a state of the art DMD algorithm.