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

2025-08-11

AI Summary

Traditional mesh-moving methods suffer from high computational complexity and poor geometric adaptability, while existing supervised learning approaches exhibit weak generalization. To address these limitations, we propose an unsupervised neural mesh adaptation framework. Our method eliminates the need for labeled training data by introducing a zero-shot generalization paradigm capable of transferring across diverse PDE types and arbitrary mesh topologies. We design an M-Uniform loss that explicitly enforces node-level equidistribution, thereby preventing mesh tangling. Furthermore, we develop a physics-informed neural architecture that jointly encodes local geometric features and physical priors, enabling end-to-end training. The approach is inherently compatible with multiscale resolutions and unstructured meshes. Experiments demonstrate significant improvements in accuracy and efficiency across various PDEs and complex geometries; errors decay stably, scalability is robust, and mesh deformations remain topologically safe.

Technology Category

• Machine Learning: Learning with Manifolds
• Game Theory and Economic Paradigms: Adversarial Learning
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Addresses high computational complexity in mesh movement techniques
• Overcomes geometric inflexibility in traditional mesh adaptation methods
• Enables zero-shot generalization across diverse PDEs and mesh topologies

Innovation

• Unsupervised mesh adaptation via geometric features
• Physics-constrained M-Uniform loss for equidistribution
• Equation-agnostic generalization across diverse PDEs