Nonparametric Stochastic Subspaces via the Bootstrap for Characterizing Model Error

2025-12-17

AI Summary

In engineering uncertainty quantification, model error—arising from the coupling of aleatory and epistemic uncertainties—is challenging to characterize accurately, thereby limiting predictive credibility. To address this, we propose a bootstrap-based nonparametric stochastic reduced-order modeling (sROM) method: within the sROM framework, the empirical distribution of state snapshot matrices is used to generate a sampling distribution of principal subspaces. This work pioneers the integration of bootstrap resampling into subspace construction—requiring no parametric assumptions, inherently satisfying physical constraints (e.g., boundary conditions), and involving only a single hyperparameter. The resulting model error characterization is unbiased, interpretable, and structure-preserving. Numerical experiments across multiple computational mechanics and structural dynamics benchmarks demonstrate substantial improvements in error representation accuracy and prediction reliability, while maintaining robustness, ease of implementation, and physical consistency.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Characterizing model error in stochastic reduced-order modeling
• Addressing aleatory and epistemic uncertainties in engineering predictions
• Improving model credibility and predictive reliability via bootstrap-based subspaces

Innovation

• Bootstrap-based stochastic subspace for model error characterization
• Assumption-free method using empirical data distribution
• Enforces linear constraints with single hyperparameter