Gaussian Processes Regression for Uncertainty Quantification: An Introductory Tutorial

2025-02-05

arXiv.org

AI Summary

Gaussian process regression (GPR) is often treated as a black-box surrogate model, limiting its pedagogical utility and interpretability in uncertainty quantification (UQ) for beginners. Core UQ tasks—including uncertainty propagation, risk estimation, Bayesian optimization, parameter inference, and sensitivity analysis—require deeper engagement with GPR’s inherent probabilistic structure. Method: This work develops a systematic, pedagogically grounded GPR-based UQ framework that integrates UQ-specific techniques: Bayesian quadrature, active learning, and surrogate-based sensitivity analysis. It emphasizes principled covariance kernel design, Bayesian hyperparameter estimation, and reproducible implementation. Contribution/Results: The framework lowers the barrier to applying GPR in complex UQ scenarios, enhances model transparency and decision reliability, and provides a theoretically rigorous yet practically actionable paradigm for UQ education and research across engineering and scientific disciplines.

Technology Category

Machine Learning: Calibration & Uncertainty QuantificationReasoning under Uncertainty: Probabilistic ProgrammingKnowledge Representation and Reasoning: Qualitative Reasoning

Application Category

Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for rankingUser Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalizationGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

Gaussian Process Regression (GPR) is a powerful nonparametric regression method that is widely used in Uncertainty Quantification (UQ) for constructing surrogate models. This tutorial serves as an introductory guide for beginners, aiming to offer a structured and accessible overview of GPR's applications in UQ. We begin with an introduction to UQ and outline its key tasks, including uncertainty propagation, risk estimation, optimization under uncertainty, parameter estimation, and sensitivity analysis. We then introduce Gaussian Processes (GPs) as a surrogate modeling technique, detailing their formulation, choice of covariance kernels, hyperparameter estimation, and active learning strategies for efficient data acquisition. The tutorial further explores how GPR can be applied to different UQ tasks, including Bayesian quadrature for uncertainty propagation, active learning-based risk estimation, Bayesian optimization for optimization under uncertainty, and surrogate-based sensitivity analysis. Throughout, we emphasize how to leverage the unique formulation of GP for these UQ tasks, rather than simply using it as a standard surrogate model.

Problem

• Introduces Gaussian Process Regression for Uncertainty Quantification tasks
• Explores GPR applications in uncertainty propagation and risk estimation
• Provides a guide for probabilistic modeling in complex computational systems

Innovation

• Gaussian Process Regression for probabilistic surrogate modeling
• Active learning strategies for efficient data acquisition
• Bayesian quadrature and optimization for uncertainty quantification