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Nonparametric Stochastic Subspaces via the Bootstrap for Characterizing Model Error

Akash Yadav, Ruda Zhang

TL;DR

This work tackles forward uncertainty quantification under epistemic and aleatory uncertainty by focusing on model-form error. It introduces SS-Bootstrap, a bootstrap-based, nonparametric stochastic subspace method within the stochastic reduced-order modeling framework, to characterize model error directly from empirical snapshots while preserving linear constraints. The approach requires only a single hyperparameter, beta, and yields sharper, yet reliable uncertainty estimates compared to Gaussian latent-variable alternatives like SS-PPCA, demonstrated on parametric linear static and linear dynamics problems with favorable computation times. The method is robust to non-Gaussian data and naturally extensible to other sampling-based model-reduction techniques and nonlinear problems, with future work aimed at model-error correction and broader MOR applications.

Abstract

Reliable forward uncertainty quantification in engineering requires methods that account for aleatory and epistemic uncertainties. In many applications, epistemic effects arising from uncertain parameters and model form dominate prediction error and strongly influence engineering decisions. Because distinguishing and representing each source separately is often infeasible, their combined effect is typically analyzed using a unified model-error framework. Model error directly affects model credibility and predictive reliability; yet its characterization remains challenging. To address this need, we introduce a bootstrap-based stochastic subspace model for characterizing model error in the stochastic reduced-order modeling framework. Given a snapshot matrix of state vectors, the method leverages the empirical data distribution to induce a sampling distribution over principal subspaces for reduced order modeling. The resulting stochastic model enables improved characterization of model error in computational mechanics compared with existing approaches. The method offers several advantages: (1) it is assumption-free and leverages the empirical data distribution; (2) it enforces linear constraints (such as boundary conditions) by construction; (3) it requires only one hyperparameter, significantly simplifying the training process; and (4) its algorithm is straightforward to implement. We evaluate the method's performance against existing approaches using numerical examples in computational mechanics and structural dynamics.

Nonparametric Stochastic Subspaces via the Bootstrap for Characterizing Model Error

TL;DR

This work tackles forward uncertainty quantification under epistemic and aleatory uncertainty by focusing on model-form error. It introduces SS-Bootstrap, a bootstrap-based, nonparametric stochastic subspace method within the stochastic reduced-order modeling framework, to characterize model error directly from empirical snapshots while preserving linear constraints. The approach requires only a single hyperparameter, beta, and yields sharper, yet reliable uncertainty estimates compared to Gaussian latent-variable alternatives like SS-PPCA, demonstrated on parametric linear static and linear dynamics problems with favorable computation times. The method is robust to non-Gaussian data and naturally extensible to other sampling-based model-reduction techniques and nonlinear problems, with future work aimed at model-error correction and broader MOR applications.

Abstract

Reliable forward uncertainty quantification in engineering requires methods that account for aleatory and epistemic uncertainties. In many applications, epistemic effects arising from uncertain parameters and model form dominate prediction error and strongly influence engineering decisions. Because distinguishing and representing each source separately is often infeasible, their combined effect is typically analyzed using a unified model-error framework. Model error directly affects model credibility and predictive reliability; yet its characterization remains challenging. To address this need, we introduce a bootstrap-based stochastic subspace model for characterizing model error in the stochastic reduced-order modeling framework. Given a snapshot matrix of state vectors, the method leverages the empirical data distribution to induce a sampling distribution over principal subspaces for reduced order modeling. The resulting stochastic model enables improved characterization of model error in computational mechanics compared with existing approaches. The method offers several advantages: (1) it is assumption-free and leverages the empirical data distribution; (2) it enforces linear constraints (such as boundary conditions) by construction; (3) it requires only one hyperparameter, significantly simplifying the training process; and (4) its algorithm is straightforward to implement. We evaluate the method's performance against existing approaches using numerical examples in computational mechanics and structural dynamics.

Paper Structure

This paper contains 20 sections, 15 equations, 13 figures, 1 table, 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 subspace model
  7. Subspace from the PCA: proper orthogonal decomposition
  8. Bootstrap: distribution-free modeling
  9. Stochastic subspace via the bootstrap
  10. Hyperparameter training
  11. Construction of SROM using SS-Bootstrap
  12. Related work
  13. Stochastic subspace via probabilistic principal component analysis (SS-PPCA)
  14. Comparison of model structures
  15. Metrics for comparing stochastic models
  16. ...and 5 more sections

Figures (13)

  • Figure 1: Dependence diagrams for (from top to bottom) the SS-PPCA and the SS-Bootstrap models 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: Linear Static Problem: prediction using Bootstrap.
  • Figure 4: Linear Static Problem: prediction using PPCA.
  • Figure 5: Linear Dynamics: space structure.
  • ...and 8 more figures