Gradient-enhancement and Gradient Predictions for Deep Gaussian Process Modeling of Expensive Computer Experiments

Abstract

Deep Gaussian processes (DGPs) are popular surrogate models for complex nonstationary computer experiments. DGPs use one or more latent Gaussian processes (GPs) to warp the input space into a plausibly stationary regime, then use typical GP regression on the warped domain. While this composition of GPs is conceptually straightforward, the functional nature of the multi-dimensional latent warping makes Bayesian posterior inference challenging. Traditional GPs with smooth kernels are naturally suited for the integration of gradient information, but the integration of gradients within a DGP presents new challenges and has yet to be explored. We propose a novel and comprehensive Bayesian framework for DGPs with gradients that facilitates both gradient-enhancement and gradient posterior predictive distributions. Our focus is on surrogate modeling of expensive, deterministic, and nonstationary computer experiments. Gradient-enhancement is most impactful when data is limited, and gradient predictions are most useful for downstream surrogate modeling tasks like optimization and active learning. We benchmark both contributions (gradient-enhanced DGPs and DGP gradient predictions), separately and together, on a variety of nonstationary test functions as well as real quantum mechanics computer experiments that simulate molecular energy and forces as a function of atomic position. On nonstationary surfaces, our gradient-enhanced DGPs outperform gradient-enhanced GPs and non-enhanced DGPs, and our DGP gradient predictions are more effective than GP gradient predictions. We provide open-source software in the "deepgp" package on CRAN, with optional Vecchia approximation to circumvent cubic computational bottlenecks.