# A Geometric Insight into Equivariant Message Passing Neural Networks on Riemannian Manifolds

arXiv: https://arxiv.org/abs/2310.10448 · HTML: https://mlsanity.com/paper/2310.10448/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 29 references that are in the map.
- First posted on arXiv 16 Oct 2023.

## Builds on (most cited first, 16 of 16)

- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [E(3)-Equivariant Graph Neural Networks for Data-Efficient and Accurate Interatomic Potentials](https://mlsanity.com/paper/2101.03164/) (2021)
- [MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields](https://mlsanity.com/paper/2206.07697/) (2022)
- [Steerable CNNs](https://mlsanity.com/paper/1612.08498/) (2016)
- [Gauge Equivariant Convolutional Networks and the Icosahedral CNN](https://mlsanity.com/paper/1902.04615/) (2019)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials](https://mlsanity.com/paper/2205.06643/) (2022)
- [Beltrami Flow and Neural Diffusion on Graphs](https://mlsanity.com/paper/2110.09443/) (2021)
- [Polar Transformer Networks](https://mlsanity.com/paper/1709.01889/) (2017)
- [Coordinate Independent Convolutional Networks -- Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds](https://mlsanity.com/paper/2106.06020/) (2021)
- [Autobahn: Automorphism-based Graph Neural Nets](https://mlsanity.com/paper/2103.01710/) (2021)
- [PELICAN: Permutation Equivariant and Lorentz Invariant or Covariant Aggregator Network for Particle Physics](https://mlsanity.com/paper/2211.00454/) (2022)
- [A General Framework for Equivariant Neural Networks on Reductive Lie Groups](https://mlsanity.com/paper/2306.00091/) (2023)
- [BIP: Boost Invariant Polynomials for Efficient Jet Tagging](https://mlsanity.com/paper/2207.08272/) (2022)
- [E(3) Equivariant Graph Neural Networks for Particle-Based Fluid Mechanics](https://mlsanity.com/paper/2304.00150/) (2023)

## Abstract (arXiv)

This work proposes a geometric insight into equivariant message passing on Riemannian manifolds. As previously proposed, numerical features on Riemannian manifolds are represented as coordinate-independent feature fields on the manifold. To any coordinate-independent feature field on a manifold comes attached an equivariant embedding of the principal bundle to the space of numerical features. We argue that the metric this embedding induces on the numerical feature space should optimally preserve the principal bundle's original metric. This optimality criterion leads to the minimization of a twisted form of the Polyakov action with respect to the graph of this embedding, yielding an equivariant diffusion process on the associated vector bundle. We obtain a message passing scheme on the manifold by discretizing the diffusion equation flow for a fixed time step. We propose a higher-order equivariant diffusion process equivalent to diffusion on the cartesian product of the base manifold. The discretization of the higher-order diffusion process on a graph yields a new general class of equivariant GNN, generalizing the ACE and MACE formalism to data on Riemannian manifolds.