Deep Gaussian Processes with Gradients

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. We provide open-source software in the "deepgp" package on CRAN, with optional Vecchia approximation to circumvent cubic computational bottlenecks. We benchmark our DGPs with gradients on a variety of nonstationary simulations, showing improvement over both GPs with gradients and conventional DGPs.