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Data-driven model order reduction for T-Product-Based dynamical systems

Shenghan Mei, Ziqin He, Yidan Mei, Xin Mao, Anqi Dong, Ren Wang, Can Chen

TL;DR

The paper tackles model order reduction for TPDSs by exploiting the $T$-product structure to avoid costly unfolding. It introduces three tensor-based MOR approaches—$T$-BT, $T$-BPOD, and $T$-ERA—each leveraging $T$-SVD and Hankel tensors to identify dominant dynamic modes in the Fourier domain. The proposed methods offer substantial memory and computational savings while maintaining comparable accuracy to traditional, unfolding-based techniques, as demonstrated on synthetic TPDSs and an image-data case study. This work has practical impact for efficiently processing high-dimensional tensor data such as image and video dynamics, enabling faster simulations, compression, and analysis without sacrificing essential dynamic characteristics.

Abstract

Model order reduction plays a crucial role in simplifying complex systems while preserving their essential dynamic characteristics, making it an invaluable tool in a wide range of applications, including robotic systems, signal processing, and fluid dynamics. However, traditional model order reduction techniques like balanced truncation are not designed to handle tensor data directly and instead require unfolding the data, which may lead to the loss of important higher-order structural information. In this article, we introduce a novel framework for data-driven model order reduction of T-product-based dynamical systems (TPDSs), which are often used to capture the evolution of third-order tensor data such as images and videos through the T-product. Specifically, we develop advanced T-product-based techniques, including T-balanced truncation, T-balanced proper orthogonal decomposition, and the T-eigensystem realization algorithm for input-output TPDSs by leveraging the unique properties of T-singular value decomposition. We demonstrate that these techniques offer significant memory and computational savings while achieving reduction errors that are comparable to those of conventional methods. The effectiveness of the proposed framework is further validated through synthetic and real-world examples.

Data-driven model order reduction for T-Product-Based dynamical systems

TL;DR

The paper tackles model order reduction for TPDSs by exploiting the -product structure to avoid costly unfolding. It introduces three tensor-based MOR approaches—-BT, -BPOD, and -ERA—each leveraging -SVD and Hankel tensors to identify dominant dynamic modes in the Fourier domain. The proposed methods offer substantial memory and computational savings while maintaining comparable accuracy to traditional, unfolding-based techniques, as demonstrated on synthetic TPDSs and an image-data case study. This work has practical impact for efficiently processing high-dimensional tensor data such as image and video dynamics, enabling faster simulations, compression, and analysis without sacrificing essential dynamic characteristics.

Abstract

Model order reduction plays a crucial role in simplifying complex systems while preserving their essential dynamic characteristics, making it an invaluable tool in a wide range of applications, including robotic systems, signal processing, and fluid dynamics. However, traditional model order reduction techniques like balanced truncation are not designed to handle tensor data directly and instead require unfolding the data, which may lead to the loss of important higher-order structural information. In this article, we introduce a novel framework for data-driven model order reduction of T-product-based dynamical systems (TPDSs), which are often used to capture the evolution of third-order tensor data such as images and videos through the T-product. Specifically, we develop advanced T-product-based techniques, including T-balanced truncation, T-balanced proper orthogonal decomposition, and the T-eigensystem realization algorithm for input-output TPDSs by leveraging the unique properties of T-singular value decomposition. We demonstrate that these techniques offer significant memory and computational savings while achieving reduction errors that are comparable to those of conventional methods. The effectiveness of the proposed framework is further validated through synthetic and real-world examples.

Paper Structure

This paper contains 19 sections, 5 theorems, 51 equations, 1 figure, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Tensors
  4. Basic T-Product Algebra
  5. T-Singular Value Decomposition
  6. Linear Model Order Reduction Techniques
  7. Balanced Truncation
  8. Balanced Proper Orthogonal Decomposition
  9. Eigensystem Realization Algorithm
  10. Data-driven Model Order Reduction for TPDSs
  11. T-Balanced Truncation
  12. T-Balanced Proper Orthogonal Decomposition
  13. T-Eigensystem Realization Algorithm
  14. Numerical Examples
  15. T-Balanced Truncation
  16. ...and 4 more sections

Key Result

Proposition 1

The input-output TPDS eq:tpds is controllable (observable) if and only if all eigentuples of the controllability Gramian $\mathscr{W}^{\text{c}}$ (observability Gramian $\mathscr{W}^{\text{o}}$) in the Fourier domain contain positive entries.

Figures (1)

  • Figure 1: (a) Impulse image data containing 21 frames. (b) Relative reconstruction errors of $\mathscr{Z}_j$ for $j=0,1,\dots,9$ using the T-ERA- and ERA-based identified systems with truncation levels $k=20,30,40,50$. The results for $k = 0$ and $k = 10$ yield similar curves to those for $k = 20$ and are omitted for clarity.

Theorems & Definitions (27)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Proposition 1
  • ...and 17 more