Gaussian process and active subspace recipe for parsimonious metamodeling

Overview & Synopsis

Numerical simulators are increasingly used to model complex physical phenomena. However, uncertainty in input parameters often limits prediction accuracy, and high computational costs of some simulators restrict the number of simulations that can be performed. This hinders critical tasks such as quantifying the uncertainties in simulation results, calibrating model parameters, and worst-case identification. A widely accepted approach is to approximate the simulator with a metamodel, based on machine learning techniques. Among these, Gaussian process regression (GPR) stands out as one of the most widely used due to its strong theoretical foundations and probabilistic uncertainty quantification capabilities. However, high-dimensional input spaces challenge GPR metamodeling, impacting both model estimation and computational efficiency in Bayesian frameworks. We propose a four-step methodology combining HSIC-based screening and Active Subspaces (AS) with GPR to construct parsimonious metamodels in challenging constraints: given-data contexts, small datasets, high-dimensional inputs spaces, and gradient-free black-box simulators. HSIC screening reduces the initial dimension to enable reliable GPR construction, while AS exploits GPR's derivative process to identify low-dimensional structure without gradient observations or optimization over embedding matrices. Key contributions include systematizing practical guidelines for covariance modeling, gradient estimation, and dimension selection, grounded in theoretical foundations linking eigenvalue spectrum to predictive quality and variance preservation. Numerical benchmarks on analytical functions with variable subset embeddings, complex random projections, and varying intrinsic dimensionalities confirm that this new algorithm has robust performance under given-data and small-data constraints and effectiveness in balancing dimensionality reduction and predictive accuracy. An industrial application about nuclear accident thermal-hydraulic simulation demonstrates 78% dimension reduction (from 27 to 6 variables) with superior prediction interval calibration.