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

Zhichao Wang, Xinhai Chen, Qinglin Wang, Xiang Gao, Qingyang Zhang, Menghan Jia, Xiang Zhang, Jie Liu

Keywords: Mesh Adaptation​, Unsupervised Learning​, Partial Differential Equations, Computational Fluid Dynamic, Zero-shot Generalization

TL;DR: This paper proposes an unsupervised mesh movement network with a novel M-Uniform loss, achieving generalization across diverse PDEs and mesh geometries without labeled data.

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 $\textbf{U}$nsupervised and $\textbf{G}$eneralizable $\textbf{M}$esh $\textbf{M}$ovement $\textbf{N}$etwork (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.

Primary Area: Machine learning for sciences (e.g. climate, health, life sciences, physics, social sciences)

Submission Number: 12196