Gaussian Process Regression for Uncertainty Quantification: An Introductory Tutorial

Abstract

Uncertainty Quantification (UQ) is essential for the reliable application of computational models in engineering and science. Among surrogate modeling techniques, Gaussian Process Regression (GPR) is particularly valuable for its non-parametric flexibility and inherent probabilistic output. This paper presents an introductory review of GPR-based methodologies within the context of 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 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. This work offers a comprehensive guide and unified framework for researchers seeking to rigorously apply probabilistic modeling to complex computational systems.