In many areas of science and engineering, computer simulations are widely used as proxies for physical experiments, which can be infeasible or unethical. Such simulations can often be computationally expensive, and an emulator can be trained to efficiently predict the desired response surface. A widely-used emulator is the Gaussian process (GP), which provides a flexible framework for efficient prediction and uncertainty quantification. Standard GPs, however, do not capture structured sparsity on the underlying response surface, which is present in many applications, particularly in the physical sciences. We thus propose a new hierarchical shrinkage GP (HierGP), which incorporates such structure via cumulative shrinkage priors within a GP framework. We show that the HierGP implicitly embeds the well-known principles of effect sparsity, heredity and hierarchy for analysis of experiments, which allows our model to identify structured sparse features from the response surface with limited data. We propose efficient posterior sampling algorithms for model training and prediction, and prove desirable consistency properties for the HierGP. Finally, we demonstrate the improved performance of HierGP over existing models, in a suite of numerical experiments and an application to dynamical system recovery.

翻译：在科学与工程的众多领域，计算机模拟常被用作物理实验的替代手段，尤其当物理实验难以实施或涉及伦理问题时。此类模拟往往计算成本高昂，因此可训练一个模拟器（emulator）以高效预测所需的响应曲面。高斯过程（Gaussian process, GP）作为一种广泛使用的模拟器，为高效预测和不确定性量化提供了灵活框架。然而，标准高斯过程无法捕捉许多应用（尤其是物理科学领域）中普遍存在的响应曲面结构化稀疏性。为此，我们提出一种新型分层收缩高斯过程（Hierarchical shrinkage GP, HierGP），通过在高斯过程框架内引入累积收缩先验（cumulative shrinkage priors）来整合此类结构化特征。我们证明，HierGP隐式嵌入了实验分析中众所周知的效应稀疏性、遗传性与层次性原则，使其能够基于有限数据从响应曲面中识别结构化稀疏特征。我们提出了用于模型训练与预测的高效后验采样算法，并证明了HierGP理想的相合性性质。最后，通过一系列数值实验及动力系统恢复应用案例，我们展示了HierGP相较于现有模型的优越性能。