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Data-driven model reduction via non-intrusive optimization of projection operators and reduced-order dynamics

Alberto Padovan, Blaine Vollmer, Daniel J. Bodony

TL;DR

This paper introduces a non-intrusive framework designed to simultaneously identify oblique projection operators and reduced-order dynamics, and shows that the gradient of the cost function with respect to the optimization parameters can be conveniently written in closed-form, so that there is no need for automatic differentiation.

Abstract

Computing reduced-order models using non-intrusive methods is particularly attractive for systems that are simulated using black-box solvers. However, obtaining accurate data-driven models can be challenging, especially if the underlying systems exhibit large-amplitude transient growth. Although these systems may evolve near a low-dimensional subspace that can be easily identified using standard techniques such as Proper Orthogonal Decomposition (POD), computing accurate models often requires projecting the state onto this subspace via a non-orthogonal projection. While appropriate oblique projection operators can be computed using intrusive techniques that leverage the form of the underlying governing equations, purely data-driven methods currently tend to achieve dimensionality reduction via orthogonal projections, and this can lead to models with poor predictive accuracy. In this paper, we address this issue by introducing a non-intrusive framework designed to simultaneously identify oblique projection operators and reduced-order dynamics. In particular, given training trajectories and assuming reduced-order dynamics of polynomial form, we fit a reduced-order model by solving an optimization problem over the product manifold of a Grassmann manifold, a Stiefel manifold, and several linear spaces (as many as the tensors that define the low-order dynamics). Furthermore, we show that the gradient of the cost function with respect to the optimization parameters can be conveniently written in closed-form, so that there is no need for automatic differentiation. We compare our formulation with state-of-the-art methods on three examples: a three-dimensional system of ordinary differential equations, the complex Ginzburg-Landau (CGL) equation, and a two-dimensional lid-driven cavity flow at Reynolds number Re = 8300.

Data-driven model reduction via non-intrusive optimization of projection operators and reduced-order dynamics

TL;DR

This paper introduces a non-intrusive framework designed to simultaneously identify oblique projection operators and reduced-order dynamics, and shows that the gradient of the cost function with respect to the optimization parameters can be conveniently written in closed-form, so that there is no need for automatic differentiation.

Abstract

Computing reduced-order models using non-intrusive methods is particularly attractive for systems that are simulated using black-box solvers. However, obtaining accurate data-driven models can be challenging, especially if the underlying systems exhibit large-amplitude transient growth. Although these systems may evolve near a low-dimensional subspace that can be easily identified using standard techniques such as Proper Orthogonal Decomposition (POD), computing accurate models often requires projecting the state onto this subspace via a non-orthogonal projection. While appropriate oblique projection operators can be computed using intrusive techniques that leverage the form of the underlying governing equations, purely data-driven methods currently tend to achieve dimensionality reduction via orthogonal projections, and this can lead to models with poor predictive accuracy. In this paper, we address this issue by introducing a non-intrusive framework designed to simultaneously identify oblique projection operators and reduced-order dynamics. In particular, given training trajectories and assuming reduced-order dynamics of polynomial form, we fit a reduced-order model by solving an optimization problem over the product manifold of a Grassmann manifold, a Stiefel manifold, and several linear spaces (as many as the tensors that define the low-order dynamics). Furthermore, we show that the gradient of the cost function with respect to the optimization parameters can be conveniently written in closed-form, so that there is no need for automatic differentiation. We compare our formulation with state-of-the-art methods on three examples: a three-dimensional system of ordinary differential equations, the complex Ginzburg-Landau (CGL) equation, and a two-dimensional lid-driven cavity flow at Reynolds number Re = 8300.
Paper Structure (7 sections, 1 theorem, 19 equations, 1 figure)

This paper contains 7 sections, 1 theorem, 19 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Mathematical formulation
  3. Petrov-Galerkin models
  4. Non-intrusive optimization of projection operators and reduced-order dynamics
  5. Gradient-based optimization on $\mathcal{M}$
  6. Connection with existing methods
  7. Application to a toy model

Key Result

Proposition 2.1

\newlabelprop:gradient0 Let eq:opt_problem be written as an equivalent unconstrained optimization problem where we seek a minimum to the Lagrangian Here, $\pmb{\lambda}_i(t) \in\mathbb{R}^r$ with $t\in [t_0,t_i]$ is the $i$th Lagrange multiplier. Defining $\mathbf{e}(t_i) \coloneqq \mathbf{y}(t_i) - \mathbf{h}\left(\pmb{\Phi}\left(\pmb{\Psi}^\intercal \pmb{\Phi}\right)^{-1}\hat{\mathbf{z}}(t_i)\

Figures (1)

  • Figure :

Theorems & Definitions (2)

  • Proposition 2.1
  • Proof 1