# Fast and Distributed Equivariant Graph Neural Networks by Virtual Node Learning

arXiv: https://arxiv.org/abs/2506.19482 · HTML: https://mlsanity.com/paper/2506.19482/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 29 of its 48 references that are in the map.
- First posted on arXiv 24 Jun 2025.

## Builds on (most cited first, 29 of 29)

- [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(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [Neural Relational Inference for Interacting Systems](https://mlsanity.com/paper/1802.04687/) (2018)
- [Learning from Protein Structure with Geometric Vector Perceptrons](https://mlsanity.com/paper/2009.01411/) (2020)
- [On Explainability of Graph Neural Networks via Subgraph Explorations](https://mlsanity.com/paper/2102.05152/) (2021)
- [Geometric and Physical Quantities Improve E(3) Equivariant Message Passing](https://mlsanity.com/paper/2110.02905/) (2021)
- [Learning Hierarchy-Aware Knowledge Graph Embeddings for Link Prediction](https://mlsanity.com/paper/1911.09419/) (2019)
- [Scalars are universal: Equivariant machine learning, structured like classical physics](https://mlsanity.com/paper/2106.06610/) (2021)
- [Artificial Intelligence for Science in Quantum, Atomistic, and Continuum Systems](https://mlsanity.com/paper/2307.08423/) (2023)
- [Conditional Antibody Design as 3D Equivariant Graph Translation](https://mlsanity.com/paper/2208.06073/) (2022)
- [Equivariant Graph Mechanics Networks with Constraints](https://mlsanity.com/paper/2203.06442/) (2022)
- [A Hitchhiker's Guide to Geometric GNNs for 3D Atomic Systems](https://mlsanity.com/paper/2312.07511/) (2023)
- [DistGNN: Scalable Distributed Training for Large-Scale Graph Neural Networks](https://mlsanity.com/paper/2104.06700/) (2021)
- [Equivariant Flows: sampling configurations for multi-body systems with symmetric energies](https://mlsanity.com/paper/1910.00753/) (2019)
- [Learning Physical Dynamics with Subequivariant Graph Neural Networks](https://mlsanity.com/paper/2210.06876/) (2022)
- [Equivariant Graph Neural Operator for Modeling 3D Dynamics](https://mlsanity.com/paper/2401.11037/) (2024)
- [Graph Classification via Deep Learning with Virtual Nodes](https://mlsanity.com/paper/1708.04357/) (2017)
- [A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications](https://mlsanity.com/paper/2403.00485/) (2024)
- [LMC: Fast Training of GNNs via Subgraph Sampling with Provable Convergence](https://mlsanity.com/paper/2302.00924/) (2023)
- [Equivariant Graph Hierarchy-Based Neural Networks](https://mlsanity.com/paper/2202.10643/) (2022)
- [Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?](https://mlsanity.com/paper/2410.11443/) (2024)
- [Duality-Induced Regularizer for Semantic Matching Knowledge Graph Embeddings](https://mlsanity.com/paper/2203.12949/) (2022)
- [Learning Rule-Induced Subgraph Representations for Inductive Relation Prediction](https://mlsanity.com/paper/2408.07088/) (2024)
- [Accurate and Scalable Graph Neural Networks via Message Invariance](https://mlsanity.com/paper/2502.19693/) (2025)

## Built on by (most cited first, 6 of 6)

- [Universally Invariant Learning in Equivariant GNNs](https://mlsanity.com/paper/2510.13169/) (2025)
- [Geometric Mixture Models for Electrolyte Conductivity Prediction](https://mlsanity.com/paper/2510.15403/) (2025)
- [IgPose: A Generative Data-Augmented Pipeline for Robust Immunoglobulin-Antigen Binding Prediction](https://mlsanity.com/paper/2603.14870/) (2026)
- [Augmented Equivariant Mesh Networks for Anatomical Segmentation](https://mlsanity.com/paper/2605.08172/) (2026)
- [LEED: Local Embedding Evolution Distance for over-smoothing estimation and virtual node selection in GNN](https://mlsanity.com/paper/2608.09596/) (2026)
- [Reducing Symmetry Increase in Equivariant Neural Networks](https://mlsanity.com/paper/2608.12010/) (2026)

## Abstract (arXiv)

Equivariant Graph Neural Networks (GNNs) have achieved remarkable success across diverse scientific applications. However, existing approaches face critical efficiency challenges when scaling to large geometric graphs and suffer significant performance degradation when the input graphs are sparsified for computational tractability. To address these limitations, we introduce FastEGNN and DistEGNN, two novel enhancements to equivariant GNNs for large-scale geometric graphs. FastEGNN employs a key innovation: a small ordered set of virtual nodes that effectively approximates the large unordered graph of real nodes. Specifically, we implement distinct message passing and aggregation mechanisms for different virtual nodes to ensure mutual distinctiveness, and minimize Maximum Mean Discrepancy (MMD) between virtual and real coordinates to achieve global distributedness. This design enables FastEGNN to maintain high accuracy while efficiently processing large-scale sparse graphs. For extremely large-scale geometric graphs, we present DistEGNN, a distributed extension where virtual nodes act as global bridges between subgraphs in different devices, maintaining consistency while dramatically reducing memory and computational overhead. We comprehensively evaluate our models across four challenging domains: N-body systems (100 nodes), protein dynamics (800 nodes), Water-3D (8,000 nodes), and our new Fluid113K benchmark (113,000 nodes). Results demonstrate superior efficiency and performance, establishing new capabilities in large-scale equivariant graph learning. Code is available at https://github.com/GLAD-RUC/DistEGNN.