Probabilistic Uncertainty Propagation Using Gaussian Process Surrogates

Abstract

This paper introduces a simple and computationally tractable probabilistic framework for forward uncertainty quantification based on Gaussian process regression, also known as Kriging. The aim is to equip uncertainty measures in the propagation of input uncertainty to simulator outputs with predictive uncertainty and confidence bounds accounting for the limited accuracy of the surrogate model, which is mainly due to using a finite amount of training data. The additional uncertainty related to the estimation of some of the prior model parameters (namely, trend coefficients and kernel variance) is further accounted for. Two different scenarios are considered. In the first one, the Gaussian process surrogate is used to emulate the actual simulator and propagate input uncertainty in the framework of a Monte Carlo analysis, i.e., as computationally cheap replacement of the original code. In the second one, semi-analytical estimates for the statistical moments of the output quantity are obtained directly based on their integral definition. The estimates for the first scenario are more general, more tractable, and they naturally extend to inputs of higher dimensions. The impact of noise on the target function is also discussed. Our findings are demonstrated based on a simple illustrative function and validated by means of several benchmark functions and a high-dimensional test case with more than a hundred uncertain variables.