# Simplicial Representation Learning with Neural k-Forms

arXiv: https://arxiv.org/abs/2312.08515 · HTML: https://mlsanity.com/paper/2312.08515/

## Key facts

- Cited by 18 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 34 references that are in the map.
- First posted on arXiv 13 Dec 2023.

## Builds on (most cited first, 23 of 23)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [How Powerful are Graph Neural Networks?](https://mlsanity.com/paper/1810.00826/) (2018)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [MoleculeNet: A Benchmark for Molecular Machine Learning](https://mlsanity.com/paper/1703.00564/) (2017)
- [Geometric deep learning: going beyond Euclidean data](https://mlsanity.com/paper/1611.08097/) (2016)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [TUDataset: A collection of benchmark datasets for learning with graphs](https://mlsanity.com/paper/2007.08663/) (2020)
- [Benchmarking Graph Neural Networks](https://mlsanity.com/paper/2003.00982/) (2020)
- [Understanding over-squashing and bottlenecks on graphs via curvature](https://mlsanity.com/paper/2111.14522/) (2021)
- [Diffusion Improves Graph Learning](https://mlsanity.com/paper/1911.05485/) (2019)
- [Weisfeiler and Lehman Go Topological: Message Passing Simplicial Networks](https://mlsanity.com/paper/2103.03212/) (2021)
- [Simplicial Neural Networks](https://mlsanity.com/paper/2010.03633/) (2020)
- [Revisiting Over-smoothing and Over-squashing Using Ollivier-Ricci Curvature](https://mlsanity.com/paper/2211.15779/) (2022)
- [Topological Graph Neural Networks](https://mlsanity.com/paper/2102.07835/) (2021)
- [Everything is Connected: Graph Neural Networks](https://mlsanity.com/paper/2301.08210/) (2023)
- [Sheaf Neural Networks](https://mlsanity.com/paper/2012.06333/) (2020)
- [Principled Simplicial Neural Networks for Trajectory Prediction](https://mlsanity.com/paper/2102.10058/) (2021)
- [Cell Complex Neural Networks](https://mlsanity.com/paper/2010.00743/) (2020)
- [Message passing all the way up](https://mlsanity.com/paper/2202.11097/) (2022)
- [Dist2Cycle: A Simplicial Neural Network for Homology Localization](https://mlsanity.com/paper/2110.15182/) (2021)
- [k-simplex2vec: a simplicial extension of node2vec](https://mlsanity.com/paper/2010.05636/) (2020)
- [Message Passing Neural Networks for Hypergraphs](https://mlsanity.com/paper/2203.16995/) (2022)

## Built on by (most cited first, 18 of 18)

- [Position: Topological Deep Learning is the New Frontier for Relational Learning](https://mlsanity.com/paper/2402.08871/) (2024)
- [E(n) Equivariant Topological Neural Networks](https://mlsanity.com/paper/2405.15429/) (2024)
- [Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures](https://mlsanity.com/paper/2407.09468/) (2024)
- [TopoTune : A Framework for Generalized Combinatorial Complex Neural Networks](https://mlsanity.com/paper/2410.06530/) (2024)
- [MANTRA: The Manifold Triangulations Assemblage](https://mlsanity.com/paper/2410.02392/) (2024)
- [Attending to Topological Spaces: The Cellular Transformer](https://mlsanity.com/paper/2405.14094/) (2024)
- [Diss-l-ECT: Dissecting Graph Data with Local Euler Characteristic Transforms](https://mlsanity.com/paper/2410.02622/) (2024)
- [Higher-Order Topological Directionality and Directed Simplicial Neural Networks](https://mlsanity.com/paper/2409.08389/) (2024)
- [Topology meets Machine Learning: An Introduction using the Euler Characteristic Transform](https://mlsanity.com/paper/2410.17760/) (2024)
- [LEAP: Local ECT-Based Learnable Positional Encodings for Graphs](https://mlsanity.com/paper/2510.00757/) (2025)
- [Cover Learning for Large-Scale Topology Representation](https://mlsanity.com/paper/2503.09767/) (2025)
- [Heat Kernel Goes Topological](https://mlsanity.com/paper/2507.12380/) (2025)
- [Directed Semi-Simplicial Learning with Applications to Brain Activity Decoding](https://mlsanity.com/paper/2505.17939/) (2025)
- [No Triangulation Without Representation: Generalization in Topological Deep Learning](https://mlsanity.com/paper/2605.06467/) (2026)
- [Neural Point-Forms](https://mlsanity.com/paper/2605.15524/) (2026)
- [Topological Signal Processing: An Application-Oriented Tutorial](https://mlsanity.com/paper/2605.22853/) (2026)
- [Cellular Sheaf Neural Operators for Structure-Preserving Surrogate Modeling of Constrained PDEs](https://mlsanity.com/paper/2606.00937/) (2026)
- [Differentiable Lifting for Topological Neural Networks](https://mlsanity.com/paper/2608.01160/) (2026)

## Abstract (arXiv)

Geometric deep learning extends deep learning to incorporate information about the geometry and topology data, especially in complex domains like graphs. Despite the popularity of message passing in this field, it has limitations such as the need for graph rewiring, ambiguity in interpreting data, and over-smoothing. In this paper, we take a different approach, focusing on leveraging geometric information from simplicial complexes embedded in ℝn using node coordinates. We use differential k-forms in ℝn to create representations of simplices, offering interpretability and geometric consistency without message passing. This approach also enables us to apply differential geometry tools and achieve universal approximation. Our method is efficient, versatile, and applicable to various input complexes, including graphs, simplicial complexes, and cell complexes. It outperforms existing message passing neural networks in harnessing information from geometrical graphs with node features serving as coordinates.