GEMove: Unsupervised Geometric Mesh Movement Beyond Density

Mesh movement adaptively relocates mesh elements under a fixed computational budget to achieve an accuracy-efficiency tradeoff in scientific simulations. Existing methods largely rely on costly theoretical supervision, whereas unsupervised approaches offer a promising alternative with zero-shot generalizability across diverse scenarios. However, under limited unsupervised constraints, local density equidistribution is prone to physically inconsistent movements and poorly coordinated over-allocation. To address these issues, we propose GEMove, which enforces global geometric equidistribution in a unified mapped space, thereby deriving vectorized movement indicators for each mesh element. Furthermore, GEMove models the mesh topology as a spring system, enabling global coordination through Tutte equilibrium. Extensive experiments across four PDE families demonstrate that GEMove achieves an average error reduction of 24.62%, with strong zero-shot generalization to unseen equations and domain geometries. Beyond simulation benchmarks, GEMove effectively adapts to real-world ocean systems, driving mesh movements in faithful response to multiscale spatial structures and long-term temporal variability. The unsupervised model learned from ocean systems further generalizes across the global Earth system, demonstrating broad scientific applicability across 54 regions and 16 variables. Code is available at https://anonymous.4open.science/r/GEMove/.