Modern scientific problems are often multi-disciplinary and require integration of computer models from different disciplines, each with distinct functional complexities, programming environments, and computation times. Linked Gaussian process (LGP) emulation tackles this challenge through a divide-and-conquer strategy that integrates Gaussian process emulators of the individual computer models in a network. However, the required stationarity of the component Gaussian process emulators within the LGP framework limits its applicability in many real-world applications. In this work, we conceptualize a network of computer models as a deep Gaussian process with partial exposure of its hidden layers. We develop a method for inference for these partially exposed deep networks that retains a key strength of the LGP framework, whereby each model can be emulated separately using a DGP and then linked together. We show in both synthetic and empirical examples that our linked deep Gaussian process emulators exhibit significantly better predictive performance than standard LGP emulators in terms of accuracy and uncertainty quantification. They also outperform single DGPs fitted to the network as a whole because they are able to integrate information from the partially exposed hidden layers. Our methods are implemented in an R package $\texttt{dgpsi}$ that is freely available on CRAN.

Gaussian processes (GPs) are widely used in computer experiments to emulate computationally expensive computer models for fast uncertainty quantification tasks such as uncertainty propagation, sensitivity analysis, and calibration. The popularity of GPs is attributed to their flexibility, native uncertainty incorporation, and analytical tractability for many key properties such as the likelihood function, predictive distribution and associated derivatives. However, GP models often assume stationarity, which in practice may not be adequate to capture non-stationary behaviors. A number of papers attempt to address this challenge by constructing non-stationary GP emulators. For example, the non-stationary covariance function introduced by Paciorek & Schervish (2003) could be adopted instead of a more standard stationary kernel. Bayesian treed Gaussian processes (TGP), proposed by Gramacy & Lee (2008), emulate non-stationary computer models by splitting the input space into several partitions and using independent stationary GPs to each sub-region. Ba et al. (2012) apply the composition of two stationary GPs to model both global and local details of a non-stationary computer model.