Vector fields are widely used to represent and model flows for many science
and engineering applications. This paper introduces a novel neural network
architecture for learning tangent vector fields that are intrinsically defined
on manifold surfaces embedded in 3D. Previous approaches to learning vector
fields on surfaces treat vectors as multi-dimensional scalar fields, using
traditional scalar-valued architectures to process channels individually, thus
fail to preserve fundamental intrinsic properties of the vector field. The core
idea of this work is to introduce a trainable vector heat diffusion module to
spatially propagate vector-valued feature data across the surface, which we
incorporate into our proposed architecture that consists of vector-valued
neurons. Our architecture is invariant to rigid motion of the input, isometric
deformation, and choice of local tangent bases, and is robust to
discretizations of the surface. We evaluate our Vector Heat Network on triangle
meshes, and empirically validate its invariant properties. We also demonstrate
the effectiveness of our method on the useful industrial application of
quadrilateral mesh generation.

介绍了一种用于在三维嵌入式曲面上学习切向量场的新型神经网络体系结构，通过引入可训练的向量热扩散模块在曲面上空间传播向量值特征数据，该架构具有刚性运动不变性、等距变形不变性，以及局部切向基的选择不变性，对曲面的离散化也具有鲁棒性。在三角网格上评估了该 Vector Heat 网络，并从实证上验证了其不变性质，同时展示了该方法在四边形网格生成方面的有用工业应用。