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

TL;DR

UGM2N is an unsupervised, physics-constrained neural network that adaptively moves mesh nodes for PDE simulations, achieving broad generalization and efficiency without pre-adapted meshes or mesh tangling issues.

Contribution

The paper introduces UGM2N, a novel unsupervised, physics-guided mesh movement network that generalizes across PDEs and mesh geometries, overcoming limitations of prior methods.

Findings

Demonstrates equation-agnostic generalization

Achieves geometric independence in mesh adaptation

Ensures error reduction without mesh tangling

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.