An Intrinsic Vector Heat Network
ICML 2024
Abstract
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.
Neural Network Architecture
Our vector-valued neural network for processing tangent vector fields utilizes vector-valued neurons with vector operations (e.g., parallel transport and the vector heat equation). Maintaining the vector nature of our data throughout results in an architecture that is invariant to isometries, rigid transformations, and the choice of tangent bases (see below). The input to our network is a set of tangent vectors defined on the vertices of a surface triangle mesh. A Vector Heat Network consists of several blocks of learned heat diffusion to harness local information, and vector MLPs to increase the expressiveness. The output is a set of tangent vector fields defined on vertices. These output fields could be regular tangent vectors or tangent vectors with N rotational symmetry defined on the tangent plane. In this section, we will illustrate individual components of our Vector Heat Network in more details.
Invariances
Our architecture possesses several fundamental invariances, which we highlight and empirically validate here, distinguishing it from the scalar-valued approach of (Dielen et al., 2021). These invariances arise from the fact that all of our operations (gradient, heat diffusion, and the per-vertex linear layer) are intrinsic, which implies that our architecture is invariant to how the mesh sits in the space.
Quadrilateral Meshing Application
To demonstrate our method’s effectiveness, we evaluate it on the task of quadrilateral remeshing. Given a triangle mesh M, we use the per-channel gradient of the first channels of the Heat Kernel Signature (HKS) as input features to our network, giving vector-valued features per vertex. The output is a 4-Rosy cross field defined on each vertex. We then interpolate the per-vertex cross field onto faces, followed by off-the-shelf algorithms by [Bommes et al., 2009] and [Ebke et al., 2013] to turn the per-face cross field into a quadrilateral mesh.
Citation
@article{gao2024vectorheatnet,
author = {Alexander Gao and Maurice Chu and Mubbsir Kapadia and Ming C. Lin and Hsueh-Ti Derek Liu},
title = {An Intrinsic Vector Heat Network},
journal = {International Conference on Machine Learning (ICML)},
year = {2024},
keywords = {Differential Geometry, Geometric Deep Learning, Graph Neural Networks, Vector Fields},
url = {https://arxiv.org/abs/2406.09648v1},
}
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