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Stochastic Subspace via Probabilistic Principal Component Analysis for Characterizing Model Error

Akash Yadav, Ruda Zhang

TL;DR

This work introduces SS-PPCA, a probabilistic subspace model based on probabilistic PCA to construct stochastic subspaces for projection-based reduced-order models. By sampling subspaces on Grassmann/Stiefel manifolds via MACG distributions and a single concentration hyperparameter $\beta$, the method yields stochastic ROMs (SROMs) that characterize model-form uncertainty with analytic tractability and minimal hyperparameter tuning. The authors propose a practical construction workflow, including snapshot collection, PCA-based subspace extraction, and a two-step hyperparameter training strategy, demonstrated across nonlinear static, static HDM-error, and dynamic space-structure problems. Results show consistent and sharp predictive intervals with modest computational cost, highlighting the approach’s potential for scalable uncertainty quantification in computational mechanics. The work also situates SS-PPCA relative to existing SROM frameworks (NPM, RSM) and discusses extensions to model-error correction in future research.

Abstract

This paper proposes a probabilistic model of subspaces based on the probabilistic principal component analysis (PCA). Given a sample of vectors in the embedding space -- commonly known as a snapshot matrix -- this method uses quantities derived from the probabilistic PCA to construct distributions of the sample matrix, as well as the principal subspaces. It is applicable to projection-based reduced-order modeling methods, such as proper orthogonal decomposition and related model reduction methods. The stochastic subspace thus constructed can be used, for example, to characterize model-form uncertainty in computational mechanics. The proposed method has multiple desirable properties: (1) it is naturally justified by the probabilistic PCA and has analytic forms for the induced random matrix models; (2) it satisfies linear constraints, such as boundary conditions of all kinds, by default; (3) it has only one hyperparameter, which significantly simplifies training; and (4) its algorithm is very easy to implement. We demonstrate the performance of the proposed method via several numerical examples in computational mechanics and structural dynamics.

Stochastic Subspace via Probabilistic Principal Component Analysis for Characterizing Model Error

TL;DR

This work introduces SS-PPCA, a probabilistic subspace model based on probabilistic PCA to construct stochastic subspaces for projection-based reduced-order models. By sampling subspaces on Grassmann/Stiefel manifolds via MACG distributions and a single concentration hyperparameter , the method yields stochastic ROMs (SROMs) that characterize model-form uncertainty with analytic tractability and minimal hyperparameter tuning. The authors propose a practical construction workflow, including snapshot collection, PCA-based subspace extraction, and a two-step hyperparameter training strategy, demonstrated across nonlinear static, static HDM-error, and dynamic space-structure problems. Results show consistent and sharp predictive intervals with modest computational cost, highlighting the approach’s potential for scalable uncertainty quantification in computational mechanics. The work also situates SS-PPCA relative to existing SROM frameworks (NPM, RSM) and discusses extensions to model-error correction in future research.

Abstract

This paper proposes a probabilistic model of subspaces based on the probabilistic principal component analysis (PCA). Given a sample of vectors in the embedding space -- commonly known as a snapshot matrix -- this method uses quantities derived from the probabilistic PCA to construct distributions of the sample matrix, as well as the principal subspaces. It is applicable to projection-based reduced-order modeling methods, such as proper orthogonal decomposition and related model reduction methods. The stochastic subspace thus constructed can be used, for example, to characterize model-form uncertainty in computational mechanics. The proposed method has multiple desirable properties: (1) it is naturally justified by the probabilistic PCA and has analytic forms for the induced random matrix models; (2) it satisfies linear constraints, such as boundary conditions of all kinds, by default; (3) it has only one hyperparameter, which significantly simplifies training; and (4) its algorithm is very easy to implement. We demonstrate the performance of the proposed method via several numerical examples in computational mechanics and structural dynamics.
Paper Structure (22 sections, 1 theorem, 23 equations, 10 figures, 1 algorithm)

This paper contains 22 sections, 1 theorem, 23 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Stochastic reduced-order modeling
  3. High-dimensional model
  4. Reduced-order model
  5. Stochastic reduced-order model
  6. Stochastic reduced-order model
  7. Stochastic subspace model
  8. Subspace from the PCA: proper orthogonal decomposition
  9. Random matrices from the PPCA
  10. Stochastic subspace via the PPCA
  11. Construction of SROM using SS-PPCA
  12. Hyperparameter training
  13. Related works
  14. Non-parametric model (NPM)
  15. Riemannian stochastic model (RSM)
  16. ...and 7 more sections

Key Result

Theorem 1

The PDF of the $\text{MACG}_{n,k}(\boldsymbol{\Sigma})$ distribution on $\text{Gr}(n, k)$ is maximized at any $k$-dim principal subspace of $\boldsymbol{\Sigma}$. If $\lambda_k > \lambda_{k+1}$, then the principal subspace $\mathscr{V}_k = \text{range}(\mathbf{V}_k)$ with $\mathbf{V}_k = [\mathbf{v}

Figures (10)

  • Figure 1: Dependence diagrams for (from top to bottom) the NPM and the RSM models of stochastic basis and the SS-PPCA model of stochastic subspace. The dashed lines indicate that the connection only exists when characterizing the ROM-to-HDM error.
  • Figure 2: HDM vs ROM displacement at the test parameter
  • Figure 3: Nonlinear static problem: prediction by SS-PPCA.
  • Figure 4: Ground truth vs HDM displacement.
  • Figure 5: Linear static problem with experimental data: SS-PPCA prediction.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof