Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Data-driven Model Reduction for Parameter-Dependent Matrix Equations via Operator Inference

Xuelian Wen, Qiuqi Li, Juan Zhang

TL;DR

The paper tackles the challenge of rapidly solving parameter-dependent matrix equations, such as PALEs and PAREs, across many parameter values. It introduces a non-intrusive, data-driven surrogate framework based on Operator Inference that learns reduced, affine-parameterized operators from solution snapshots while preserving the polynomial structure via vectorization. By combining POD-based basis reduction with regression on projected data, the method achieves accurate reduced representations without assembling or storing full-order operators, enabling fast online evaluation and scalable many-query analyses. Empirical results across continuous-time and discrete-time PALEs, continuous-time coupled PALEs, and PAREs demonstrate substantial speedups (often orders of magnitude) with competitive or superior accuracy, validating the approach as a practical tool for parametric matrix equations. The work highlights the importance of representative training data and suggests future extensions to broader nonlinear matrix equations and alternative data-driven reduction strategies.

Abstract

This work develops a non-intrusive, data-driven surrogate modeling framework based on Operator Inference (OpInf) for rapidly solving parameter-dependent matrix equations in many-query settings. Motivated by the requirements of the OpInf methodology, we reformulate the matrix equations into a structured representation that explicitly shows the parameter dependence in polynomial form. This reformulation is crucial for efficient model reduction. This approach constructs reduced-order models via regression on solution snapshots, bypassing the need for expensive full-order operators and thus overcoming the primary bottlenecks of intrusive methods in high-dimensional contexts. Numerical experiments confirm their accuracy and computational efficiency, demonstrating that our work is a scalable and practical solution for parameter-dependent matrix equations.

Data-driven Model Reduction for Parameter-Dependent Matrix Equations via Operator Inference

TL;DR

The paper tackles the challenge of rapidly solving parameter-dependent matrix equations, such as PALEs and PAREs, across many parameter values. It introduces a non-intrusive, data-driven surrogate framework based on Operator Inference that learns reduced, affine-parameterized operators from solution snapshots while preserving the polynomial structure via vectorization. By combining POD-based basis reduction with regression on projected data, the method achieves accurate reduced representations without assembling or storing full-order operators, enabling fast online evaluation and scalable many-query analyses. Empirical results across continuous-time and discrete-time PALEs, continuous-time coupled PALEs, and PAREs demonstrate substantial speedups (often orders of magnitude) with competitive or superior accuracy, validating the approach as a practical tool for parametric matrix equations. The work highlights the importance of representative training data and suggests future extensions to broader nonlinear matrix equations and alternative data-driven reduction strategies.

Abstract

This work develops a non-intrusive, data-driven surrogate modeling framework based on Operator Inference (OpInf) for rapidly solving parameter-dependent matrix equations in many-query settings. Motivated by the requirements of the OpInf methodology, we reformulate the matrix equations into a structured representation that explicitly shows the parameter dependence in polynomial form. This reformulation is crucial for efficient model reduction. This approach constructs reduced-order models via regression on solution snapshots, bypassing the need for expensive full-order operators and thus overcoming the primary bottlenecks of intrusive methods in high-dimensional contexts. Numerical experiments confirm their accuracy and computational efficiency, demonstrating that our work is a scalable and practical solution for parameter-dependent matrix equations.

Paper Structure

This paper contains 12 sections, 4 theorems, 62 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. The Operator Inference Framework
  3. OpInf for parameter-dependent matrix equations
  4. Vectorization of parameter-dependence matrix equations
  5. Reduced-order basis construction
  6. Non-intrusive Operator Inference
  7. Numerical experiments
  8. Continuous-time PALEs
  9. Discrete-time PALEs
  10. Continuous-time coupled PALEs
  11. Continuous-time PAREs
  12. Summary

Key Result

Lemma 3.1

Let $X\in\mathbb{R}^{n\times n}$ be a symmetric matrix, and let $e_i$ denote the $i$-th column of the identity matrix $I_n$. Then, the vectorization of the quadratic term $XGX$ satisfies where $G=[g_{ij}]\in\mathbb{R}^{n\times n},\ x=\operatorname{vec}(X),$ and $E_{ij}=(\boldsymbol{e}_j^\top\otimes I_n)\otimes (\boldsymbol{e}_i^\top\otimes I_n)\in\mathbb{R}^{n^2\times n^4}.$ Furthermore, owing to

Figures (11)

  • Figure 1: Relative errors for different basis selection methods (left) and for the POD method with varying size of training parameters (right), evaluated at the full-order system dimension of $N=1024^2$ over $10^4$ uniformly distributed random test parameters.
  • Figure 2: The snapshots, evaluated at the full-order system dimension of $N=1024^2$.
  • Figure 3: The POD modes, evaluated at the full-order system dimension of $N=1024^2$.
  • Figure 4: Distribution of the 200 training parameters (left) and the average relative error of the model trained on them, evaluated at the full-order system dimension of $N=512^2$ over $1000$ uniformly distributed random test parameters (right).
  • Figure 5: Distribution of the 50 training parameters (left) and the average relative error of the model trained on them, evaluated at the full-order system dimension of $N=512^2$ over $1000$ uniformly distributed random test parameters (right).
  • ...and 6 more figures

Theorems & Definitions (10)

  • Lemma 3.1: Vectorization of quadratic term
  • Remark 3.2
  • Theorem 3.3: Energy-based POD truncation
  • Proof 1
  • Remark 3.4
  • Theorem 3.5
  • Proof 2
  • Theorem 3.6: Existence and Uniqueness of the Reduced-Order Operators
  • Proof 3
  • Remark 3.7