# Surface Networks

## TL;DR

Surface Networks (SNs) enhance 3D mesh representations by leveraging extrinsic geometry, specifically the Dirac operator, to improve deformation stability and modeling power in shape analysis tasks.

## Contribution

The paper introduces Surface Networks that incorporate the Dirac operator for extrinsic geometry, surpassing Laplacian-based models in stability and expressiveness.

## Key findings

- SNs are stable to deformation and discretization.
- SNs outperform existing models in mesh deformation prediction.
- SNs enable effective generative modeling of 3D surfaces.

## Abstract

We study data-driven representations for three-dimensional triangle meshes, which are one of the prevalent objects used to represent 3D geometry. Recent works have developed models that exploit the intrinsic geometry of manifolds and graphs, namely the Graph Neural Networks (GNNs) and its spectral variants, which learn from the local metric tensor via the Laplacian operator. Despite offering excellent sample complexity and built-in invariances, intrinsic geometry alone is invariant to isometric deformations, making it unsuitable for many applications. To overcome this limitation, we propose several upgrades to GNNs to leverage extrinsic differential geometry properties of three-dimensional surfaces, increasing its modeling power. In particular, we propose to exploit the Dirac operator, whose spectrum detects principal curvature directions --- this is in stark contrast with the classical Laplace operator, which directly measures mean curvature. We coin the resulting models \emph{Surface Networks (SN)}. We prove that these models define shape representations that are stable to deformation and to discretization, and we demonstrate the efficiency and versatility of SNs on two challenging tasks: temporal prediction of mesh deformations under non-linear dynamics and generative models using a variational autoencoder framework with encoders/decoders given by SNs.