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, with solutions over the spatial domain. However, obtaining these solutions are often prohibitively costly, limiting the feasibility of exploring parameters in PDEs. In this paper, we propose an efficient emulator that simultaneously predicts the solutions over the spatial domain, 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 Gaussian process models with the same hyperparameters in each of them. Most importantly, by revealing the underlying clustering structures, the proposed method can provide valuable insights into qualitative features of the resulting dynamics 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 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）在空间域上求解。然而，获取这些解的代价往往过高，限制了对偏微分方程参数探索的可行性。本文提出一种高效的模拟器，可同时预测空间域上的解，并对其不确定性量化提供理论支撑。所提方法的新颖之处在于将网格节点坐标融入统计模型：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具有相同超参数的高斯过程模型。最重要的是，通过揭示潜在的聚类结构，该方法可为所得动力学定性特征提供重要见解，从而指导后续研究。实际算例表明，与主要竞争方法相比，本文方法在预测误差更小的同时具有相当的运算时间，并能识别出具有物理意义（如满足边界条件）的网格节点聚类。本方法配套的R语言程序包已在开源仓库中提供。