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化方法（如有限元法）在空间域上求解，然而获取这些解的代价往往过高，限制了探索偏微分方程参数的可行性。本文提出一种高效代理模型，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持。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，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。更重要的是，通过揭示潜在的聚类结构，该方法能够为系统动力学的定性特征提供有价值的见解，从而指导后续研究。实际案例表明，与主要竞争方法相比，本方法在保持竞争性计算时间的同时，可显著降低预测误差，并识别出具有物理意义（如满足边界条件）的网格节点聚类。本文方法对应的R语言开源软件包已发布于公共代码仓库。