Deep Gaussian Process Emulation with gradient Information and Sequential Design for Simulators with Sharp Variations

Abstract

Deep Gaussian Processes (DGPs) compose GP layers to warp inputs, enabling improved emulation of computer models with nonstationary input-output behavior compared with ordinary GPs. In contrast to GPs, the predictive uncertainty for DGP gradients remains relatively underexplored. Quantifying DGP gradient uncertainty can support gradient-based tasks in complex, nonstationary settings where ordinary GPs may struggle. While GP gradient posteriors are analytically tractable, extending such constructions to DGPs is challenging due to their hierarchical composition. In this paper, we propose an efficient approximation to the gradient distribution of a two-layer DGP emulator. Using the chain rule with local linearization, we derive closed-form expressions for the gradient mean and covariance, enabling fast gradient evaluation with uncertainty quantification (UQ). Empirically, our approach delivers promising performance while uniquely providing UQ of gradients. We then use the gradient uncertainties to guide sequential design for models with sharp variations: we define sharp variation regions as those where the gradient norm exceeds a threshold. We subsequently introduce an entropy-based acquisition rule that selects new samples in locations where the classification of points as inside versus outside the sharp-variation region is most uncertain. Experiments on synthetic benchmarks and a real-world application show that the resulting sequential design more accurately emulates functions with sharp variations than existing design methods.