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Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approach

Jin Yi Yong, Rudy Geelen, Johann Guilleminot

TL;DR

A probabilistic approach to represent and quantify model-form uncertainties in the reduced-order modeling of complex systems using operator inference techniques by combining Riemannian projection and retraction operators - acting on a subset of the Stiefel manifold - with an information-theoretic formulation.

Abstract

This paper presents a probabilistic approach to represent and quantify model-form uncertainties in the reduced-order modeling of complex systems using operator inference techniques. Such uncertainties can arise in the selection of an appropriate state-space representation, in the projection step that underlies many reduced-order modeling methods, or as a byproduct of considerations made during training, to name a few. Following previous works in the literature, the proposed method captures these uncertainties by expanding the approximation space through the randomization of the projection matrix. This is achieved by combining Riemannian projection and retraction operators - acting on a subset of the Stiefel manifold - with an information-theoretic formulation. The efficacy of the approach is assessed on canonical problems in fluid mechanics by identifying and quantifying the impact of model-form uncertainties on the inferred operators.

Learning Latent Space Dynamics with Model-Form Uncertainties: A Stochastic Reduced-Order Modeling Approach

TL;DR

A probabilistic approach to represent and quantify model-form uncertainties in the reduced-order modeling of complex systems using operator inference techniques by combining Riemannian projection and retraction operators - acting on a subset of the Stiefel manifold - with an information-theoretic formulation.

Abstract

This paper presents a probabilistic approach to represent and quantify model-form uncertainties in the reduced-order modeling of complex systems using operator inference techniques. Such uncertainties can arise in the selection of an appropriate state-space representation, in the projection step that underlies many reduced-order modeling methods, or as a byproduct of considerations made during training, to name a few. Following previous works in the literature, the proposed method captures these uncertainties by expanding the approximation space through the randomization of the projection matrix. This is achieved by combining Riemannian projection and retraction operators - acting on a subset of the Stiefel manifold - with an information-theoretic formulation. The efficacy of the approach is assessed on canonical problems in fluid mechanics by identifying and quantifying the impact of model-form uncertainties on the inferred operators.
Paper Structure (21 sections, 44 equations, 17 figures, 2 algorithms)

This paper contains 21 sections, 44 equations, 17 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Formulation for Learning Latent Space Dynamics on Nonlinear Manifolds
  3. Learning Nonlinear Representations
  4. Learning Latent Space Dynamics
  5. Probabilistic Formulation for Model-Form Uncertainties
  6. Stochastic Representation of the Projection Matrix
  7. Preliminaries
  8. Probabilistic Representation on a Subset of the Stiefel Manifold
  9. On Constructing the Global Projection Matrix
  10. Parameterization of the Probabilistic Model
  11. Selection of Reduced-Order Matrix Operators for Uncertainty Propagation
  12. Summary of the Proposed Approach
  13. Numerical Results
  14. Application to Burgers' Equation
  15. Problem Description
  16. ...and 6 more sections

Figures (17)

  • Figure 1: Overview of the stochastic reduced-order modeling approach operating in the Operator Inference (OpInf) framework. Model-form uncertainties are generated using a randomization of the projection matrix, and parameters in the reduced-order approximation are selected on the fly to propagate uncertainties to quantities of interest.
  • Figure 2: Full-order solution of the Burgers' equation \ref{['eqn: burgers']} at various time instances for different values of the initial condition parameter $\mu$. The horizontal axis ($x \in \Omega$) represents the spatial domain, while the vertical axis ($s$) denotes the scalar-valued time-dependent velocity.
  • Figure 3: Parametric analysis of the reduced-order model obtained via the OpInf method for the Burgers' equation \ref{['eqn: burgers']}. The horizontal axis represents the number of reduced basis vectors $r$ in $[V]$, which corresponds to $\epsilon(r)$ values of $1\times 10^{-1}$, $5\times 10^{-2}$, $1 \times 10^{-2}$, $5 \times 10^{-3}$, and $1 \times 10^{-3}$. (a) Relative state error \ref{['eqn: relativerror']} as a function $r$ with $q=2$ basis vectors in $[\overline{V}]$ for MPOD-OpInfPoly. (b) Relative state error as a function of $r$ for POD-OpInf.
  • Figure 4: Relative state error \ref{['eqn: relativerror']} as a function of the number of reduced basis vectors $q$ in $[\overline{V}]$ for MPOD-OpInf. The horizontal axis denotes $q$ number of reduced basis vectors in $[\overline{V}]$.
  • Figure 5: Graph of the error function $r \mapsto \epsilon(r)$ for different combinations of parameter $\mu$. The vertical axis represents the error function $\epsilon(r)$ as defined in Eq. \ref{['eqn: errorfunction']}. The horizontal axis shows the rank $r$ of the projection matrices $[\Phi^{(i)}]$, where $1 \leq i \leq n_c$, and $n_c$ is the number of parameter combinations. Distinct colors indicate different combination groups $^9C_k$ (where $3\leq k \leq 9$ and $^nC_k$ denotes $k$-combinations from a set of $n$ elements). The number of combinations within each group is denoted by $n$ in the figure.
  • ...and 12 more figures

Theorems & Definitions (2)

  • Remark 1
  • Remark 2