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

Jie Liu (City University of Hong Kong) · Xiang Zhang (University of California, San Diego) · Zhichao Wang (University of California, Berkeley) · Qingyang Zhang (Tianjin University Tencent AI Lab) · Xiang Gao (, Tsinghua University) · Xinhai Chen (National University of Defense Technology) · Qinglin Wang (National University of Defense Technology) · Menghan Jia (National University of Defense Technology)

computational complexity computational efficiencyequation-agnostic generalizationgeometric inflexibilitylocalized geometric feature learningm-uniform lossmesh movement techniquesmulti-scale resolutionsnumerical solutionspartial differential equationsphysics-constrained loss functionsimulation accuracy supervised learningunsupervised mesh adaptationzero-shot generalization

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.