An Intrinsic Vector Heat Network

TL;DR

The paper tackles learning tangent vector fields on manifolds embedded in $\mathbb{R}^3$ by introducing an intrinsic Vector Heat Network that preserves vector properties through vector-valued features and a trainable vector heat diffusion module. It builds on the vector heat diffusion framework with a deterministic connection Laplacian, enabling invariance to rigid motion, isometry, and local tangent-basis choices, while remaining robust to surface discretization. A key contribution is the use of spectral acceleration for diffusion and the extension to $N$-Rosy fields, facilitating vector-field-based quadrilateral remeshing as demonstrated on avatar heads and animal models, with strong generalization and ablation results. The proposed approach advances geometric deep learning on meshes by delivering a principled, intrinsic method for vector-field processing that generalizes across mesh geometries and supports practical applications like quad meshing and robotic navigation on curved terrains.

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.

Figures (19)

• Figure 1: The tangent plane $T_p \mathcal{M}$ at point $p$ on a manifold $\mathcal{M}$ does not have a canonical choice of basis vectors ${\mathbf{e}}_0, {\mathbf{e}}_1$. Our proposed architecture for learning tangent vector fields is invariant to choice of tangent bases.
• Figure 2: Our vector heat network is a neural network of complex-valued neurons complexNNsurvey with (1) a Vector Heat Diffusion module (see \ref{['sec:learned_diffusion']} ) and (2) a vector MLP module (see \ref{['sec:complex_linear']} ). Starting with a Vector MLP to transform input features from $\mathbb{C}^{n \times c^\text{in}}$ to $\mathbb{C}^{n \times c^l}$, our method consists of several layers of the Vector Heat Diffusion (red) and Vector MLP (blue) with skip connections, followed by another Vector MLP to map the feature to output dimensions.
• Figure 3: The vector heat diffusion process presented in \ref{['equ:implicit_heat_diffusion']} smears out a tangent vector field ${\mathbf{u}}_t$ to its neighbors to obtain another tangent vector field ${\mathbf{u}}_{t+1}$.
• Figure 4: N-Rosy fields refer to tangent vectors that are N-way rotationally symmetricPalaciosZ07rosy. For instance, $N=1$ refers to the usual 2D tangent vector, $N=2$ to a straight line, and $N=4$ to a "cross" field.
• Figure 5: Our architecture is invariant to rigid transformation. A model trained on a mesh at one orientation (1st) generalizes to its rigidly transformed counterpart (2nd), outputting a tangent vector field with no error (3rd). This differs from the baseline method of dielen2021learning, which outputs a different vector field (4th) with high error (5th).