[Paper Review] The Gaussian process modelling module in UQLab

Christos Lataniotis, Stefano Marelli+1 more|arXiv (Cornell University)|Sep 27, 2017

Probabilistic and Robust Engineering Design33 references9 citations

TL;DR

This paper presents the Gaussian process (GP) modelling module in UQLab, a flexible, open-source MATLAB toolbox for Kriging-based surrogate modelling in uncertainty quantification. It enables seamless integration into UQ workflows, supports custom correlation functions, and demonstrates performance across engineering applications including wind turbine simulations and geotechnical problems with non-stationary correlations.

ABSTRACT

We introduce the Gaussian process (GP) modelling module developed within the UQLab software framework. The novel design of the GP-module aims at providing seamless integration of GP modelling into any uncertainty quantification workflow, as well as a stand-alone surrogate modelling tool. We first briefly present the key mathematical tools at the basis of GP modelling (a.k.a. Kriging), as well as the associated theoretical and computational framework. We then provide an extensive overview of the available features of the software and demonstrate its flexibility and user-friendliness. Finally, we showcase the usage and the performance of the software on several applications borrowed from different fields of engineering. These include a basic surrogate of a well-known analytical benchmark function, a hierarchical Kriging example applied to wind turbine aero-servo-elastic simulations and a more complex geotechnical example that requires a non-stationary, user-defined correlation function. The GP-module, like the rest of the scientific code that is shipped with UQLab, is open source (BSD license).

Motivation & Objective

• To develop a general-purpose, user-friendly Gaussian process modelling module within the UQLab framework for uncertainty quantification (UQ) workflows.
• To enable seamless integration of Kriging surrogate models into diverse UQ applications, including reliability analysis, sensitivity analysis, and optimization.
• To support advanced customization, such as user-defined, non-stationary correlation functions, for complex engineering problems.
• To provide a flexible, extensible software solution that supports both novice users and researchers requiring fine-grained control over hyperparameters and model structure.
• To address limitations in existing toolboxes by offering full configurability, extensibility, and compatibility with broader UQ workflows.

Proposed method

• The GP module implements Kriging (Gaussian process regression) using a hierarchical, object-oriented architecture within the UQLab framework.
• It supports both stationary and non-stationary correlation functions, with user-defined correlation functions implemented via custom MATLAB function handles.
• Hyperparameter optimization is performed using external MATLAB toolboxes, with support for bounds and initial values matching the number of hyperparameters.
• The module computes predictive means and variances, enabling uncertainty quantification and adaptive sampling strategies.
• It integrates with UQLab’s existing modules, such as Polynomial Chaos-Kriging and reliability analysis, for end-to-end UQ pipelines.
• The implementation uses the built-in function `uq_Kriging_eval_R` to evaluate correlation matrices efficiently, even across multiple regions with different correlation structures.

Experimental results

Research questions

• RQ1 How can a general-purpose, extensible Gaussian process module be designed to support diverse uncertainty quantification workflows in engineering?
• RQ2 To what extent can user-defined, non-stationary correlation functions be effectively integrated into a Kriging surrogate model within a software framework?
• RQ3 How does the GP module perform in real-world engineering applications involving complex, high-dimensional, or noisy simulation data?
• RQ4 Can the module support hierarchical Kriging models for multi-fidelity or multi-scale simulations?
• RQ5 What is the impact of custom correlation function design on surrogate model accuracy and computational efficiency?

Key findings

• The GP module successfully enables accurate surrogate modelling of a benchmark analytical function with minimal user input, demonstrating ease of use.
• In the wind turbine aero-servo-elastic simulation, the hierarchical Kriging model achieved high accuracy with adaptive sampling, validating its use in complex simulation workflows.
• For the geotechnical inverse problem, the custom non-stationary correlation function—based on crack location and orientation—improved model fit compared to standard stationary models.
• The length scale estimates showed higher error in the direction with fewer samples, highlighting the importance of experimental design in Kriging.
• The module supports complex correlation functions with multiple hyperparameters, as demonstrated by the vectorized, readable implementation of region-dependent correlation structures.
• The software is open-source under the BSD license and has been actively used in UQLab’s ecosystem, with future support for unknown noise levels and improved optimization integration planned.