Constructing confidence and prediction intervals for multifidelity surrogate models involving noisy data

Abstract

Nowadays, computer simulations, or white-box models, are indispensable to model complex engineering systems that need to be reliable and safe. White-box models can provide accurate predictions when there is a precise underlying physical model, but they may be hard to obtain for highly complex engineering systems, and they often fail to capture reality in its entirety. Sometimes, experimental data are available for the same system. Such data can be used to construct black-box models, and while these models are flexible, they do not necessarily honour the underlying physical reality [3]. When both experimental data and one or more white-box computational models are available, it is natural to look for an approach that combines the white- and black-box modelling paradigms to benefit from the strengths of both. This combined approach is referred to as grey-box modelling. Grey-box modelling is concerned with the problem of merging information from data-driven black-box models and white-box models. In this work, as part of the Marie Skłodowska-Curie Innovative Training Network GREYDIENT, we propose to perform this task by using multifidelity (MF) surrogate modelling. A MF surrogate model combines information from models at different computational fidelities into a new surrogate model, typically supplementing a very small high-fidelity data set with larger lower-fidelity ones [1, 2]. More precisely, we consider the experimental data as single realisations of black-box high-fidelity models, whereas the available white-box computational models are considered as their low-fidelity counterparts. In our MF setting, uncertainty is expected in both the high- and the low-fidelity models. Specifically, in the case of the high-fidelity experimental data, it takes the form of aleatory uncertainty, and it is due to the inevitable measurement noise resulting from the limited precision and resolution of measurement devices. Moreover, what we consider as low-fidelity computational models are expensive-to-evaluate white-box models, and therefore, we substitute them using surrogate models. As the available experimental design for the construction of the latter is relatively small, due to computational budget constraints, we consider their predictions as affected by epistemic uncertainty. This work proposes a theoretical and methodological framework for MF surrogate modelling that considers deterministic models contaminated by noise. This framework first provides predictions for the underlying noise-free high-fidelity model, or in other words, it acts as a denoiser for noisy high-fidelity data. Secondly, it quantifies the different kinds of uncertainty in its predictions by estimating confidence intervals for the mean prediction and prediction intervals for future high-fidelity observations. Our methodology is applied and validated on a variety of analytical use cases. References [1] M. C. Kennedy and A. O’Hagan. Predicting the output from a complex computer code when fast approximations are available. Biometrika, 87(1):1– 13, 2000. [2] L. Le Gratiet and J. Garnier. Recursive co-Kriging model for design of computer experiments with multiple levels of fidelity. International Journal for Uncertainty Quantification, 4(5):365–386, 2014 [3] H. J. A. F. Tulleken. Grey-box modelling and identification using physical knowledge and Bayesian techniques. Automatica, 29(2):285–308, 1993.