Surrogate models for uncertainty quantification in computational sciences

Abstract

Nowadays, computational models are used in virtually all fields of applied sciences and engineering to predict the behaviour of complex natural or man-made systems. These models, a.k.a. simulators allow engineers to assess the performance of a system in-silico, and then optimize its design or operating. Simulators such as high-fidelity finite element models usually feature dozens of parameters and are costly to run, even when taking full advantage of the available computer power. In parallel, the more complex the system, the more uncertainty in its governing parameters, environmental and operating conditions. In this respect, uncertainty quantification (UQ) methods used to solve reliability, sensitivity or optimal design problems have gained interest in both academia and the industry in the last decade. Monte Carlo simulation is a well-known, brute-force method based on random number generation to solve these problems. It usually requires thousands to millions of simulations for accurate predictions though, which is not tractable with high-fidelity simulators. In contrast, surrogate models allow us to tackle these UQ problems by constructing an accurate approximation of the simulator’s response from a limited number of runs at selected values (the so-called experimental design) and a learning algorithm. In this lecture, general features of surrogate models and their link with machine learning will be first introduced. Polynomial chaos expansions will then be discussed in details, together with their sparse version for high-dimensional problems. Recent extensions to dynamics will be addressed and applications in sensitivity will be shown as an illustration. Gaussian processes and their use together with active learning will also be addressed to solve reliability analysis.