Universally Invariant Learning in Equivariant GNNs

2025/10/15 by Jiacheng Cen, Anyi Li, Cen, Jiacheng +13 · 2 citations
Computer Science · #FOS: Computer and information sciences #Machine Learning (cs.LG) #Neural Networks and Applications

paper · pdf · doi:10.48550/arxiv.2510.13169

openalex publication_date 2025/10/15 · openalex created_date 2025/10/17 · openalex updated_date 2026/07/28

Abstract

Equivariant Graph Neural Networks (GNNs) have demonstrated significant success across various applications. To achieve completeness -- that is, the universal approximation property over the space of equivariant functions -- the network must effectively capture the intricate multi-body interactions among different nodes. Prior methods attain this via deeper architectures, augmented body orders, or increased degrees of steerable features, often at high computational cost and without polynomial-time solutions. In this work, we present a theoretically grounded framework for constructing complete equivariant GNNs that is both efficient and practical. We prove that a complete equivariant GNN can be achieved through two key components: 1) a complete scalar function, referred to as the canonical form of the geometric graph; and 2) a full-rank steerable basis set. Leveraging this finding, we propose an efficient algorithm for constructing complete equivariant GNNs based on two common models: EGNN and TFN. Empirical results demonstrate that our model demonstrates superior completeness and excellent performance with only a few layers, thereby significantly reducing computational overhead while maintaining strong practical efficacy.

Citations

• Geometric Mixture Models for Electrolyte Conductivity Prediction
• Fast and Distributed Equivariant Graph Neural Networks by Virtual Node Learning
• The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products
• GeoRecon: Graph-Level Representation Learning for 3D Molecules via Reconstruction-Based Pretraining
• DualEquiNet: A Dual-Space Hierarchical Equivariant Network for Large Biomolecules
• Equivariant Spherical Transformer for Efficient Molecular Modeling
• Tokenizing Electron Cloud in Protein-Ligand Interaction Learning
• PolyConf: Unlocking Polymer Conformation Generation through Hierarchical Generative Models
• Siamese Foundation Models for Crystal Structure Prediction
• SE(3)-Equivariant Ternary Complex Prediction Towards Target Protein Degradation
• A Survey of Graph Transformers: Architectures, Theories and Applications
• ReQFlow: Rectified Quaternion Flow for Efficient and High-Quality Protein Backbone Generation
• Large Language-Geometry Model: When LLM meets Equivariance
• E2Former: An Efficient and Equivariant Transformer with Linear-Scaling Tensor Products
• Geometric Trajectory Diffusion Models
• Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?
• WGFormer: An SE(3)-Transformer Driven by Wasserstein Gradient Flows for Molecular Ground-State Conformation Prediction
• Geometric Representation Condition Improves Equivariant Molecule Generation
• Powder Diffraction Crystal Structure Determination Using Generative Models
• S-MolSearch: 3D Semi-supervised Contrastive Learning for Bioactive Molecule Search
• Force-Guided Bridge Matching for Full-Atom Time-Coarsened Dynamics of Peptides
• On the Expressive Power of Sparse Geometric MPNNs
• MMPolymer: A Multimodal Multitask Pretraining Framework for Polymer Property Prediction
• E(n) Equivariant Topological Neural Networks
• A Survey of Geometric Graph Neural Networks: Data Structures, Models and Applications
• Full-Atom Peptide Design with Geometric Latent Diffusion
• On the Completeness of Invariant Geometric Deep Learning Models
• Weisfeiler Leman for Euclidean Equivariant Machine Learning
• SEGNO: Generalizing Equivariant Graph Neural Networks with Physical Inductive Biases
• Crystal Structure Prediction by Joint Equivariant Diffusion
• De novo design of protein structure and function with RFdiffusion
• EquiformerV2: Improved Equivariant Transformer for Scaling to Higher-Degree Representations
• Generalist Equivariant Transformer Towards 3D Molecular Interaction Learning
• FAENet: Frame Averaging Equivariant GNN for Materials Modeling
• Recognizing Rigid Patterns of Unlabeled Point Clouds by Complete and Continuous Isometry Invariants with no False Negatives and no False Positives
• Three iterations of (d-1)-WL test distinguish non isometric clouds of d-dimensional points
• EquiPocket: an E(3)-Equivariant Geometric Graph Neural Network for Ligand Binding Site Prediction
• Is Distance Matrix Enough for Geometric Deep Learning?
• End-to-End Full-Atom Antibody Design
• Equivariance with Learned Canonicalization Functions
• Conditional Antibody Design as 3D Equivariant Graph Translation
• Energy-Motivated Equivariant Pretraining for 3D Molecular Graphs
• e3nn: Euclidean Neural Networks
• Equiformer: Equivariant Graph Attention Transformer for 3D Atomistic Graphs
• ComENet: Towards Complete and Efficient Message Passing for 3D Molecular Graphs
• MACE: Higher Order Equivariant Message Passing Neural Networks for Fast and Accurate Force Fields
• Equivariant Graph Mechanics Networks with Constraints
• Roto-translated Local Coordinate Frames For Interacting Dynamical Systems
• Frame Averaging for Invariant and Equivariant Network Design
• Geometric and Physical Quantities Improve E(3) Equivariant Message Passing
• Molecule Generation by Principal Subgraph Mining and Assembling
• GemNet: Universal Directional Graph Neural Networks for Molecules
• E(n) Equivariant Graph Neural Networks
• Spherical Message Passing for 3D Graph Networks
• Equivariant message passing for the prediction of tensorial properties and molecular spectra
• On the Universality of Rotation Equivariant Point Cloud Networks
• Learning from Protein Structure with Geometric Vector Perceptrons
• Directional Message Passing for Molecular Graphs
• Atomic Cluster Expansion: Completeness, Efficiency and Stability
• Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds
• Spherical CNNs
• SchNet – A deep learning architecture for molecules and materials
• Deep Sets
• Practical graph isomorphism, II

Cited by

• Geometric Mixture Models for Electrolyte Conductivity Prediction
• Identifiable Equivariant Networks are Layerwise Equivariant

Related

• Are High-Degree Representations Really Unnecessary in Equivariant Graph Neural Networks?
• Universal Invariant and Equivariant Graph Neural Networks
• On Universality Classes of Equivariant Networks
• E(n) Equivariant Graph Neural Networks
• A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks