Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, the outputs of which consist of the solutions on a set of mesh nodes over the spatial domain. However, these simulations are often prohibitively costly to survey the input space. In this paper, we propose an efficient emulator that simultaneously predicts the outputs on a set of mesh nodes, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits a Gaussian process model in each. Most importantly, by revealing the underlying clustering structures, the proposed method can extract valuable flow physics present in the systems that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that exhibit coherent input-output relationships and possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.

翻译：偏微分方程已成为模拟复杂物理系统的重要工具。这类方程通常通过网格方法（如有限元方法）进行数值求解，其输出由空间域内一组网格节点上的解组成。然而，此类仿真在探索输入空间时往往计算成本过高。本文提出了一种高效的仿真器，可同时预测一组网格节点上的输出，并从理论上证明了其不确定性量化的合理性。该方法的核心创新在于将网格节点坐标纳入统计模型。具体而言，该方法通过狄利克雷过程先验将网格节点划分为多个聚类，并在每个聚类中拟合高斯过程模型。最重要的是，通过揭示潜在的聚类结构，该方法可提取系统中存在的有价值流动物理特性，为后续研究提供指导。实际案例表明，与主要竞争方法相比，我们的方法具有更小的预测误差和具有竞争力的计算时间，并能识别出具有一致输入-输出关系且具备物理意义（如满足边界条件）的有趣网格节点聚类。本方法对应的R语言软件包已发布于开源存储库中。