Comprehensive Robust Dynamic Mode Decomposition from Mode Extraction to Dimensional Reduction

TL;DR

This paper introduces CR-DMD, a comprehensive framework that enhances the robustness of Dynamic Mode Decomposition against mixed noise, improving mode extraction and low-dimensional modeling accuracy in noisy dynamical systems.

Contribution

It presents a novel convex optimization-based preprocessing and a new convex formulation for dimensional reduction, ensuring stable and faithful low-dimensional representations under noise.

Findings

CR-DMD outperforms existing robust DMD methods in accuracy.

It effectively removes mixed noise from fluid dynamics data.

The proposed method maintains high fidelity in low-dimensional models.

Abstract

We propose Comprehensive Robust Dynamic Mode Decomposition (CR-DMD), a novel framework that robustifies the entire DMD process - from mode extraction to dimensional reduction - against mixed noise. Although standard DMD widely used for uncovering spatio-temporal patterns and constructing low-dimensional models of dynamical systems, it suffers from significant performance degradation under noise due to its reliance on least-squares estimation for computing the linear time evolution operator. Existing robust variants typically modify the least-squares formulation, but they remain unstable and fail to ensure faithful low-dimensional representations. First, we introduce a convex optimization-based preprocessing method designed to effectively remove mixed noise, achieving accurate and stable mode extraction. Second, we propose a new convex formulation for dimensional reduction that explicitly links the robustly extracted modes to the original noisy observations, constructing a faithful representation of the original data via a sparse weighted sum of the modes. Both stages are efficiently solved by a preconditioned primal-dual splitting method. Experiments on fluid dynamics datasets demonstrate that CR-DMD consistently outperforms state-of-the-art robust DMD methods in terms of mode accuracy and fidelity of low-dimensional representations under noisy conditions.