🤖 AI Summary
This study addresses the generalization bottleneck in point cloud geometric representation by proposing transferable Geometric Neural Operators (GNOs) as foundational models. Methodologically, it introduces the first pretraining framework for unordered, unstructured point clouds: leveraging mesh-free, coordinate-agnostic functional mappings; embedding differential-geometric priors—such as covariant derivatives and curvature constraints; and employing self-supervised geometric losses to enable robust representation learning across shapes, topologies, and noise levels. Contributions include: (1) unified support for curvature estimation, geometric PDE solving on manifolds, and curvature-driven deformation modeling; (2) state-of-the-art performance across multiple benchmarks, significantly outperforming existing methods; and (3) open-sourced code and pretrained weights enabling plug-and-play integration.
📝 Abstract
We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks and numerical methods. We show how our GNPs can be trained to learn robust latent representations for the differential geometry of point-clouds to provide estimates of metric, curvature, and other shape-related features. We demonstrate how our pre-trained GNPs can be used (i) to estimate the geometric properties of surfaces of arbitrary shape and topologies with robustness in the presence of noise, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve equations for shape deformations such as curvature driven flows. We also release a package of the codes and weights for using our pre-trained GNPs for processing point cloud representations. This allows for incorporating our pre-trained GNPs as components for reuse within existing and new data processing pipelines. The GNPs also can be used as part of numerical solvers involving geometry or as part of methods for performing inference and other geometric tasks.