Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A polynomial approximation scheme for nonlinear model reduction by moment matching

Carlos Doebeli, Alessandro Astolfi, Dante Kalise, Alessio Moreschini, Giordano Scarciotti, Joel Simard

TL;DR

This work tackles the challenge of nonlinear moment matching for high-dimensional dynamical systems by approximating the invariance PDE that defines the centre manifold with a Galerkin spectral expansion and Newton iterations on coefficient vectors. The method yields a practical reduced-order model whose steady-state response matches the full system under a given signal generator, even when the full model has thousands of states and non-polynomial nonlinearities. Key contributions include the first explicit computational framework for nonlinear moment matching in this setting, demonstrations up to $n=1000$, and a tensor-enabled assembly that mitigates the curse of dimensionality. The approach offers a scalable route to accurate ROMs for complex interconnected systems and opens directions for higher-dimensional signal generators and broader control applications.

Abstract

We propose a procedure for the numerical approximation of invariance equations arising in the moment matching technique associated with reduced-order modeling of high-dimensional dynamical systems. The Galerkin residual method is employed to find an approximate solution to the invariance equation using a Newton iteration on the coefficients of a monomial basis expansion of the solution. These solutions to the invariance equations can then be used to construct reduced-order models. We assess the ability of the method to solve the invariance PDE system as well as to achieve moment matching and recover a system's steady-state behaviour for linear and nonlinear signal generators with system dynamics up to $n=1000$ dimensions.

A polynomial approximation scheme for nonlinear model reduction by moment matching

TL;DR

This work tackles the challenge of nonlinear moment matching for high-dimensional dynamical systems by approximating the invariance PDE that defines the centre manifold with a Galerkin spectral expansion and Newton iterations on coefficient vectors. The method yields a practical reduced-order model whose steady-state response matches the full system under a given signal generator, even when the full model has thousands of states and non-polynomial nonlinearities. Key contributions include the first explicit computational framework for nonlinear moment matching in this setting, demonstrations up to , and a tensor-enabled assembly that mitigates the curse of dimensionality. The approach offers a scalable route to accurate ROMs for complex interconnected systems and opens directions for higher-dimensional signal generators and broader control applications.

Abstract

We propose a procedure for the numerical approximation of invariance equations arising in the moment matching technique associated with reduced-order modeling of high-dimensional dynamical systems. The Galerkin residual method is employed to find an approximate solution to the invariance equation using a Newton iteration on the coefficients of a monomial basis expansion of the solution. These solutions to the invariance equations can then be used to construct reduced-order models. We assess the ability of the method to solve the invariance PDE system as well as to achieve moment matching and recover a system's steady-state behaviour for linear and nonlinear signal generators with system dynamics up to dimensions.

Paper Structure

This paper contains 19 sections, 1 theorem, 62 equations, 3 figures, 16 tables.

Table of Contents

  1. Introduction
  2. Contribution
  3. Organization
  4. Moment Matching for Nonlinear Dynamical Systems
  5. Notion of Moment
  6. Moment-Based Reduced-Order Modeling
  7. Approximate solutions via Galerkin expansion
  8. Polynomial expansion of the invariant mapping
  9. Polynomial basis
  10. Integration
  11. Computational complexity and implementation
  12. Evaluating approximate solutions
  13. Numerical results
  14. Summary of examples
  15. Test 1: Low-dimensional nonlinear system
  16. ...and 4 more sections

Key Result

Proposition 2.6

\newlabelprop0 If Assumptions ass:stab and ass:f hold, then we are guaranteed the existence of a mapping $\pi : \mathbb{R}^d \to \mathbb{R}^n$ that is defined on a neighbourhood $W^0$ about the origin and characterizes a locally attractive centre manifold for eq:interconnected. This mapping $\pi$ This centre manifold is represented as $\mathcal{M} = \{(x, \omega) : x = \pi(\omega)\}$ for $\pi$ s

Figures (3)

  • Figure 1: Diagrammatic illustration of the concepts of moment matching in reduced-order modeling. The signal generator generates a control input to the higher order model and the reduced-order model, and moment-matching corresponds to the exact matching of their steady-state responses.
  • Figure 1: Plot of $y(t)$ and $y_r(t)$ for $t \in [0, 50]$ for the RL ladder problem with a linear oscillator signal generator for a domain $\Omega = [-1,1]^2$ and a maximum degree $M = 6$ for the dimension $n = 1000$, with constants $a = 2, \kappa = 1.1, c = 10$ and initial conditions $\omega_0 = (0.1, 0.2)^\top$, $r_0 = (0,1)^\top$ and $x_0 = 0$.
  • Figure 2: Plot of $y(t)$ and $y_r(t)$ for $t \in [0, 50]$ for the RL ladder problem with a Van der Pol oscillator signal generator for a domain $\Omega = [-1,1]^2$ and a maximum degree $M = 4$ for the dimension $n = 1000$, with constants $\mu = 0.25, \kappa = 1.1, c = 10$ and initial conditions $\omega_0 = (0.1, 0.2)^\top$, $r_0 = (0,1)^\top$ and $x_0 = 0$.

Theorems & Definitions (11)

  • Definition 2.1: Observability
  • Definition 2.2: Accessibility
  • Definition 2.3: Poisson stability
  • Definition 2.4: Neutral stability
  • Definition 2.5: Centre Manifold, carr
  • Proposition 2.6
  • Definition 2.7: Moment
  • Definition 2.8
  • Definition 3.1
  • Remark 3.2
  • ...and 1 more