Gaussian Processes Regression for Uncertainty Quantification: An Introductory Tutorial (2502.03090v2)

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.

• The paper provides an introductory tutorial on applying Gaussian Process Regression (GPR) for Uncertainty Quantification (UQ) across various tasks.
• GPR serves as a nonparametric Bayesian surrogate model, uniquely providing explicit uncertainty estimates through its probabilistic formulation and kernel functions.
• GPR is applied effectively in core UQ tasks such as Uncertainty Propagation, Risk Estimation, and Sensitivity Analysis, offering advantages for complex models and limited data.

Gaussian Processes Regression for Uncertainty Quantification

This paper provides an introductory tutorial on the application of Gaussian Process Regression (GPR) to Uncertainty Quantification (UQ). Authored by Jinglai Li and Hongqiao Wang, it serves as a structured guide for beginners exploring GPR's role in various UQ tasks. This essay will present a concise overview of the paper, emphasizing its core contributions and implications for researchers in the field.

Overview of Uncertainty Quantification

The paper begins by discussing the importance of UQ in assessing and managing uncertainty inherent in computer models. These uncertainties arise due to aleatory factors, such as inherent randomness, and epistemic factors, which result from a lack of knowledge. The inability to eliminate aleatory uncertainty contrasts with the potential reduction of epistemic uncertainty through data acquisition. When both types coexist, distinguishing them becomes challenging, and the focus is on addressing the overall uncertainty.

Key UQ Tasks

The authors define several UQ tasks where GPR can be applied:

• Uncertainty Propagation (UP): Quantifying how input uncertainties affect output predictions.
• Risk Estimation (RE): Assessing the likelihood of adverse events, often described by failure probabilities.
• Optimization under Uncertainty (OU): Finding optimal decisions in the presence of uncertainty using stochastic optimization techniques.
• Parameter Estimation (PE): Estimating parameters from data within a Bayesian framework to capture estimation uncertainty.
• Sensitivity Analysis (SA): Understanding which model inputs have significant effects on outputs.

GPR as a Surrogate Modeling Technique

GPR, a nonparametric Bayesian method, is highlighted for its strength in providing uncertainty estimates through its probabilistic formulation. GPR represents functions as distributions over functions, described via covariance kernels. These provide explicit posterior predictive distributions, allowing for uncertainty quantification in predictions. The selection of appropriate covariance kernels is crucial, with popular choices including the Squared Exponential and Matérn kernels, each characterized by hyperparameters estimated through Maximum Likelihood Estimation (MLE) or Cross-Validation (CV).

Applications in UQ Tasks

1. Uncertainty Propagation: GPR substitutes for the true model in Monte Carlo simulations, enabling efficient calculation of statistical quantities like means and variances.
2. Risk Estimation: GPR is used in combination with active learning strategies, such as the Active Kriging Monte Carlo Simulation (AK-MCS), to improve estimates of failure probabilities.
3. Optimization under Uncertainty: Bayesian Optimization (BO), a key method using GPR, balances exploration and exploitation in search processes to manage constraints and maximize objectives.
4. Parameter Estimation: GPR assists in approximating the log likelihood in Bayesian PE, optimizing sample efficiency through informed data acquisition.
5. Sensitivity Analysis: GPR facilitates the computation of sensitivity indices by reducing computational demands typical of Monte Carlo-based methods.

Implications and Future Developments

GPR's integration in UQ tasks highlights its flexibility and strengths, especially in situations of limited data and high computational cost. The paper discusses GPR's advantages, such as its nonparametric nature and UQ capacity, alongside limitations like managing high-dimensional inputs and scalability challenges. Its potential to complement deep neural networks (DNNs) is also noted, where GPR's uncertainty quantification offers a meaningful addition to DNN’s predictive power.

In conclusion, this paper underscores GPR's value as a surrogate modeling framework within UQ contexts. By amalgamating theoretical insights with practical methodologies, it lays a foundation for further research into enhancing GPR's applicability and efficiency across diverse scientific and engineering domains. Researchers are encouraged to explore the potential of GPR in complex systems modeling and consider its integration with advanced learning algorithms for comprehensive UQ undertaking.