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

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 .