Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Tensor-based homogeneous polynomial dynamical system analysis from data

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

TL;DR

The paper tackles the challenge of analyzing and identifying homogeneous polynomial dynamical systems (HPDSs) with higher-order interactions by introducing a tensor-decomposition framework that leverages tensor train (TTD) and hierarchical Tucker (HTD) representations. It develops memory-efficient, data-driven methods for estimating HPDS parameters directly from time-series data, including autonomous and input–output settings, and derives necessary and sufficient conditions for controllability and observability that exploit low-rank tensor structures. The authors demonstrate substantial reductions in memory and computational complexity compared to full-tensor approaches and validate the framework with numerical experiments across various tensor-generation schemes. This work significantly advances practical HPDS analysis, enabling scalable identification and system-theoretic analysis for high-dimensional, higher-order networks common in biology, ecology, and engineering.

Abstract

Numerous complex real-world systems, such as those in biological, ecological, and social networks, exhibit higher-order interactions that are often modeled using polynomial dynamical systems or homogeneous polynomial dynamical systems (HPDSs). However, identifying system parameters and analyzing key system-theoretic properties remain challenging due to their inherent nonlinearity and complexity, particularly for large-scale systems. To address these challenges, we develop an innovative computational framework in this article that leverages advanced tensor decomposition techniques, namely tensor train and hierarchical Tucker decompositions, to facilitate efficient identification and analysis of HPDSs that can be equivalently represented by tensors. Specifically, we introduce memory-efficient system identification techniques for directly estimating system parameters represented through tensor decompositions from time-series data. Additionally, we develop necessary and sufficient conditions for determining controllability and observability using the tensor decomposition-based representations of HPDSs, accompanied by detailed complexity analyses that demonstrate significant reductions in computational demands. The effectiveness and efficiency of our framework are validated through numerical examples.

Tensor-based homogeneous polynomial dynamical system analysis from data

TL;DR

The paper tackles the challenge of analyzing and identifying homogeneous polynomial dynamical systems (HPDSs) with higher-order interactions by introducing a tensor-decomposition framework that leverages tensor train (TTD) and hierarchical Tucker (HTD) representations. It develops memory-efficient, data-driven methods for estimating HPDS parameters directly from time-series data, including autonomous and input–output settings, and derives necessary and sufficient conditions for controllability and observability that exploit low-rank tensor structures. The authors demonstrate substantial reductions in memory and computational complexity compared to full-tensor approaches and validate the framework with numerical experiments across various tensor-generation schemes. This work significantly advances practical HPDS analysis, enabling scalable identification and system-theoretic analysis for high-dimensional, higher-order networks common in biology, ecology, and engineering.

Abstract

Numerous complex real-world systems, such as those in biological, ecological, and social networks, exhibit higher-order interactions that are often modeled using polynomial dynamical systems or homogeneous polynomial dynamical systems (HPDSs). However, identifying system parameters and analyzing key system-theoretic properties remain challenging due to their inherent nonlinearity and complexity, particularly for large-scale systems. To address these challenges, we develop an innovative computational framework in this article that leverages advanced tensor decomposition techniques, namely tensor train and hierarchical Tucker decompositions, to facilitate efficient identification and analysis of HPDSs that can be equivalently represented by tensors. Specifically, we introduce memory-efficient system identification techniques for directly estimating system parameters represented through tensor decompositions from time-series data. Additionally, we develop necessary and sufficient conditions for determining controllability and observability using the tensor decomposition-based representations of HPDSs, accompanied by detailed complexity analyses that demonstrate significant reductions in computational demands. The effectiveness and efficiency of our framework are validated through numerical examples.

Paper Structure

This paper contains 20 sections, 10 theorems, 55 equations, 3 figures, 6 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Kronecker product
  4. Tensors
  5. Tensor products
  6. Tensor unfolding
  7. Tensor decomposition
  8. Tensor-based HPDSs
  9. System identification for tensor-based HPDSs
  10. Identification of autonomous system
  11. Identification of TTD- and HTD-based HPDSs
  12. Identification with input and output data
  13. Tensor-based HPDS analysis
  14. Controllability
  15. Obeservability
  16. ...and 5 more sections

Key Result

Proposition 3.1

\newlabelthm: ausysid0 The data $(\textbf{X}_0, \textbf{X}_1)$ is sufficient for unique identification of the tensor-based HPDS (eq: ausys) if and only if

Figures (3)

  • Figure 1: An example of the TTD of a third-order tensor.
  • Figure 1: Comparison of memory consumption for the full, TTD-based, and HTD-based representations of tensor-based HPDSs across different tensor orders. The three panels, from left to right, represent symmetric tensor generation, low TT-rank tensor generation, and low hierarchical-rank tensor generation, respectively.
  • Figure 2: Computational time comparison for full, TTD-based, and HTD-based representations of tensor-based HPDSs across varying tensor orders and dimensions. The panels in the first row display tensors generated with symmetry, the second row shows tensors generated with low TT-ranks, and the third row presents tensors generated with low hierarchical ranks.

Theorems & Definitions (29)

  • Proposition 3.1
  • Proof 1
  • Proposition 3.2
  • Proof 2
  • Remark 3.3
  • Remark 3.4
  • Proposition 3.5
  • Proof 3
  • Proposition 3.6
  • Proof 4
  • ...and 19 more