Deep Gaussian Processes (DGPs), multi-layered extensions of GPs, better emulate simulators with regime transitions or sharp changes than standard GPs. Gradient information is crucial for tasks like sensitivity analysis and dimension reduction. Although gradient posteriors are well-defined in GPs, extending them to DGPs is challenging due to their hierarchical structure. We propose a novel method to approximate the DGP emulator's gradient distribution, enabling efficient gradient computation with uncertainty quantification (UQ). Our approach derives an analytical gradient mean and the covariance. The numerical results show that our method outperforms GP and DGP with finite difference methods in gradient accuracy, offering the extra unique benefit of UQ. Based on the gradient information, we further propose a sequential design criterion to identify the sharp variation regions efficiently, with the gradient norm as a key indicator whose distribution can be readily evaluated in our framework. We evaluated the proposed sequential design using synthetic examples and empirical applications, demonstrating its superior performance in emulating functions with sharp changes compared to existing design methods. The DGP gradient computation is seamlessly integrated into the advanced Python package dgpsi for DGP emulation, along with the proposed sequential design available at https://github.com/yyimingucl/DGP.

翻译：深度高斯过程（DGPs）作为高斯过程（GPs）的多层扩展，在模拟具有机制转换或急剧变化的模拟器时优于标准GPs。梯度信息对于敏感性分析和降维等任务至关重要。尽管梯度后验在GPs中已有明确定义，但由于其层次结构，将其扩展到DGPs具有挑战性。我们提出了一种新颖的方法来近似DGP模拟器的梯度分布，从而实现具有不确定性量化（UQ）的高效梯度计算。我们的方法推导了梯度的解析均值和协方差。数值结果表明，我们的方法在梯度精度上优于使用有限差分法的GP和DGP，并额外提供了UQ这一独特优势。基于梯度信息，我们进一步提出了一种序贯设计准则，以梯度范数作为关键指标（其分布可在我们的框架中轻松评估），从而高效识别急剧变化区域。我们通过合成示例和实证应用评估了所提出的序贯设计，结果表明其在模拟具有急剧变化的函数方面优于现有设计方法。DGP梯度计算已无缝集成到用于DGP模拟的高级Python软件包dgpsi中，所提出的序贯设计亦可在https://github.com/yyimingucl/DGP获取。