A Complete Guide to Spherical Equivariant Graph Transformers

TL;DR

This paper provides a comprehensive, mathematically detailed guide to spherical equivariant graph neural networks, explaining their theoretical foundations, architectures, and applications in molecular and biomolecular modeling.

Contribution

It offers an intuitive, self-contained introduction to spherical equivariant modeling, including derivations, code snippets, and practical architectures like Tensor Field Network and SE(3)-Transformer.

Findings

Develops a complete mathematical foundation for spherical EGNNs

Constructs and explains key architectures like Tensor Field Network and SE(3)-Transformer

Provides practical guidance for implementation in molecular applications

Abstract

Spherical equivariant graph neural networks (EGNNs) provide a principled framework for learning on three-dimensional molecular and biomolecular systems, where predictions must respect the rotational symmetries inherent in physics. These models extend traditional message-passing GNNs and Transformers by representing node and edge features as spherical tensors that transform under irreducible representations of the rotation group SO(3), ensuring that predictions change in physically meaningful ways under rotations of the input. This guide develops a complete, intuitive foundation for spherical equivariant modeling - from group representations and spherical harmonics, to tensor products, Clebsch-Gordan decomposition, and the construction of SO(3)-equivariant kernels. Building on this foundation, we construct the Tensor Field Network and SE(3)-Transformer architectures and explain how they perform equivariant message-passing and attention on geometric graphs. Through clear mathematical derivations and annotated code excerpts, this guide serves as a self-contained introduction for researchers and learners seeking to understand or implement spherical EGNNs for applications in chemistry, molecular property prediction, protein structure modeling, and generative modeling.