UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss

TL;DR

UGM2N addresses the need for fast, generalizable mesh movement without pre-adapted meshes by introducing node-patch-based, unsupervised learning and a physics-informed M-Uniform loss that enforces local mesh equidistribution in a metric space. The method uses a graph Transformer to predict patch-wise node updates, incorporating Hessian-based flow features and iterating over epochs with interpolation to maintain consistency. Key contributions include a theoretically grounded loss that bounds local variance to approximate equidistribution, demonstrated equation-agnostic generalization across Poisson, Helmholtz, Burgers, and wave-like problems, and robust performance across varied mesh geometries with favorable scalability and no mesh tangling. The approach offers significant practical impact by delivering efficient, generalizable mesh adaptation suitable for real-time or large-scale simulations across diverse PDEs and geometries.

Abstract

Partial differential equations (PDEs) form the mathematical foundation for modeling physical systems in science and engineering, where numerical solutions demand rigorous accuracy-efficiency tradeoffs. Mesh movement techniques address this challenge by dynamically relocating mesh nodes to rapidly-varying regions, enhancing both simulation accuracy and computational efficiency. However, traditional approaches suffer from high computational complexity and geometric inflexibility, limiting their applicability, and existing supervised learning-based approaches face challenges in zero-shot generalization across diverse PDEs and mesh topologies.In this paper, we present an Unsupervised and Generalizable Mesh Movement Network (UGM2N). We first introduce unsupervised mesh adaptation through localized geometric feature learning, eliminating the dependency on pre-adapted meshes. We then develop a physics-constrained loss function, M-Uniform loss, that enforces mesh equidistribution at the nodal level.Experimental results demonstrate that the proposed network exhibits equation-agnostic generalization and geometric independence in efficient mesh adaptation. It demonstrates consistent superiority over existing methods, including robust performance across diverse PDEs and mesh geometries, scalability to multi-scale resolutions and guaranteed error reduction without mesh tangling.

Figures (20)

• Figure 1: The proposed mesh movement network.
• Figure 2: Mesh adaptation results for the Helmholtz equation with different solutions.
• Figure 2: The mesh adaptation performance of models trained with different loss functions
• Figure 3: The ER for different mesh resolutions on the Helmholtz equation. For clarity, we clipped the minimum ER at -1%, even though for some methods (e.g., M2N), their adapted meshes significantly increased the solution error.
• Figure 4: The performance of mesh movement methods under different mesh resolutions is presented. The dashed line in the figure indicates the time required for UGM2N to complete a single iteration.