A hybrid polynomial chaos expansion – Gaussian process regression method for Bayesian uncertainty quantification and sensitivity analysis

Abstract

This paper introduces a novel hybrid method for uncertainty quantification (UQ) combining the benefits of polynomial chaos expansion (PCE) and Gaussian process regression (GPR). The proposed method features a GPR formulation that leverages special implicit kernels involving an infinite sequence of some of the orthogonal polynomials from the Wiener-Askey scheme. These kernels enable the closed-form calculation of PCE coefficients by analytical integration of the GPR posterior, thereby leading to a Bayesian estimation in terms of both expected value and covariance matrix. Notably, the Bayesian definition allows associating confidence information to the computed coefficients, which is then propagated to the classical closed-form estimates associated to PCEs, i.e., first- and second-order moments and Sobol’ sensitivity indices. The advocated method helps mitigate some long-standing shortcomings of PCEs in terms of training efficiency and scalability to higher dimensions, while providing an accurate quantification of the intrinsic model uncertainty. Moreover, it allows for a nonparametric computation of PCE coefficients, since the basis functions do not need to be selected a priori. A simple and effective multi-output formulation, involving the tuning of a single set of hyperparameters, is also discussed. The hybrid PCE-GPR method is extensively illustrated and validated based on both synthetic examples and high-dimensional application test cases in the field of electrical engineering, for which it is shown to substantially outperform state-of-the-art PCE methods such as least-angle regression, orthogonal matching pursuit, subspace pursuit, and Bayesian compressive sensing.