Mesh-clustered Gaussian process emulator for partial differential equation boundary value problems

TL;DR

This paper introduces a novel Gaussian process emulator for PDE boundary value problems that leverages mesh node clustering to improve prediction accuracy and provide physical insights, with theoretical uncertainty quantification.

Contribution

The method uniquely incorporates mesh node clustering via Dirichlet process priors into Gaussian process models, enhancing PDE solution emulation and interpretability.

Findings

Smaller prediction errors compared to competitors

Provides meaningful physical clusters of mesh nodes

Maintains competitive computational efficiency

Abstract

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.