# Surface Networks

arXiv: https://arxiv.org/abs/1705.10819 · HTML: https://mlsanity.com/paper/1705.10819/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 May 2017.

## Built on by (most cited first, 14 of 14)

- [Subdivision-Based Mesh Convolution Networks](https://mlsanity.com/paper/2106.02285/) (2021)
- [Understanding and Improving Features Learned in Deep Functional Maps](https://mlsanity.com/paper/2303.16527/) (2023)
- [Frame Averaging for Equivariant Shape Space Learning](https://mlsanity.com/paper/2112.01741/) (2021)
- [DeltaConv: Anisotropic Operators for Geometric Deep Learning on Point Clouds](https://mlsanity.com/paper/2111.08799/) (2021)
- [Stability of Neural Networks on Manifolds to Relative Perturbations](https://mlsanity.com/paper/2110.04702/) (2021)
- [Convergence of Invariant Graph Networks](https://mlsanity.com/paper/2201.10129/) (2022)
- [Generalizable Local Feature Pre-training for Deformable Shape Analysis](https://mlsanity.com/paper/2303.15104/) (2023)
- [Dense Graph Convolutional Neural Networks on 3D Meshes for 3D Object Segmentation and Classification](https://mlsanity.com/paper/2106.15778/) (2021)
- [Graph Neural Networks in Computer Vision -- Architectures, Datasets and Common Approaches](https://mlsanity.com/paper/2212.10207/) (2022)
- [MeT: A Graph Transformer for Semantic Segmentation of 3D Meshes](https://mlsanity.com/paper/2307.01115/) (2023)
- [Modeling Dynamics over Meshes with Gauge Equivariant Nonlinear Message Passing](https://mlsanity.com/paper/2310.19589/) (2023)
- [HodgeFormer: Transformers for Learnable Operators on Triangular Meshes through Data-Driven Hodge Matrices](https://mlsanity.com/paper/2509.01839/) (2025)
- [A Recipe for Geometry-Aware 3D Mesh Transformers](https://mlsanity.com/paper/2411.00164/) (2024)
- [SVarM: Linear Support Varifold Machines for Classification and Regression on Geometric Data](https://mlsanity.com/paper/2506.01189/) (2025)

## Abstract (arXiv)

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 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.