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Comprehensive Robust Dynamic Mode Decomposition from Mode Extraction to Dimensional Reduction

Yuki Nakamura, Shingo Takemoto, Shunsuke Ono

TL;DR

CR-DMD tackles the robustness problem of Dynamic Mode Decomposition under mixed noise by proposing a two-stage convex framework. The first stage performs a TV-based, convex denoising to enable stable mode extraction, while the second stage crafts a faithful, low-dimensional representation by explicitly linking the robust modes to the original observations via a convex dimensional-reduction problem solved with Preconditioned Primal-Dual Splitting. The approach yields high-accuracy mode extraction and superior low-rank reconstructions on fluid-flow datasets, outperforming RPCA, TLS-DMD, OptDMD, CDMD, and PiDMD, especially under strong noise and outliers. This framework, with its data-fidelity constraints and automatic stepsize selection, offers practical robustness and scalability for spatiotemporal data analysis in noisy environments, and paves the way for integrating convex robustness into operator-centric DMD variants.

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.

Comprehensive Robust Dynamic Mode Decomposition from Mode Extraction to Dimensional Reduction

TL;DR

CR-DMD tackles the robustness problem of Dynamic Mode Decomposition under mixed noise by proposing a two-stage convex framework. The first stage performs a TV-based, convex denoising to enable stable mode extraction, while the second stage crafts a faithful, low-dimensional representation by explicitly linking the robust modes to the original observations via a convex dimensional-reduction problem solved with Preconditioned Primal-Dual Splitting. The approach yields high-accuracy mode extraction and superior low-rank reconstructions on fluid-flow datasets, outperforming RPCA, TLS-DMD, OptDMD, CDMD, and PiDMD, especially under strong noise and outliers. This framework, with its data-fidelity constraints and automatic stepsize selection, offers practical robustness and scalability for spatiotemporal data analysis in noisy environments, and paves the way for integrating convex robustness into operator-centric DMD variants.

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.
Paper Structure (24 sections, 42 equations, 7 figures, 4 tables, 2 algorithms)

This paper contains 24 sections, 42 equations, 7 figures, 4 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations and Definitions
  4. Dynamic Mode decomposition (DMD)
  5. Proximal Tools
  6. Preconditioned Primal-Dual Splitting Method (P-PDS)
  7. Proposed Method
  8. Preprocessing
  9. Problem Formulation
  10. Optimization Algorithm
  11. Dimensional Reduction
  12. Problem Formulation
  13. Optimization Algorithm
  14. Comparison with Existing Methods
  15. Reasonable Preprocessing
  16. ...and 9 more sections

Figures (7)

  • Figure 1: Schematic illustration of the proposed CR-DMD framework.
  • Figure 2: Estimated dynamics results for the noisy cylinder flow. The target modes indicate the ground truth dynamics associated with each mode for comparison. The results are arranged by noise condition in rows and by the eigenvalue corresponding to each mode in columns. In each plot, the 95% confidence ellipses and the mean predictions for each method are displayed.
  • Figure 3: Estimated dynamics results for the noisy turbulent channel flow. The target modes indicate the ground truth dynamics associated with each mode for comparison. The results are arranged by noise condition in rows and by the eigenvalue corresponding to each mode in columns. In each plot, the 95% confidence ellipses and the mean predictions for each method are displayed.
  • Figure 4: Dimensional reduction results for Cylinder Wake with 5 modes (the 1st snapshot). In each subfigure, the upper image is the reconstructed vorticity field, and the lower image is the absolute difference between the ground truth and the reconstructed image. The magnified region in the bottom-right corner highlights detailed vortex structures.
  • Figure 5: Dimensional reduction results for Channel Flow with 21 modes (except OptDMD using 39 modes for stability) (the $10$-th snapshot). In each subfigure, the upper image is the reconstructed vorticity field, and the lower image is the absolute difference between the ground truth and the reconstructed image. The magnified region in the bottom-right corner highlights detailed vortex structures.
  • ...and 2 more figures