A Clifford Algebraic Approach to E(n)-Equivariant High-order Graph Neural Networks

TL;DR

This paper introduces CG-EGNNs, a novel Clifford algebra-based high-order equivariant graph neural network that enhances expressive power and captures geometric symmetries, outperforming previous methods in physical and chemical data benchmarks.

Contribution

It proposes CG-EGNNs, integrating Clifford algebras with high-order message passing to improve equivariance and expressive power in geometric graph neural networks.

Findings

CG-EGNNs outperform previous methods on benchmark datasets.

The universality of k-hop message passing is established.

Enhanced model performance in physical and chemical applications.

Abstract

Designing neural network architectures that can handle data symmetry is crucial. This is especially important for geometric graphs whose properties are equivariance under Euclidean transformations. Current equivariant graph neural networks (EGNNs), particularly those using message passing, have a limitation in expressive power. Recent high-order graph neural networks can overcome this limitation, yet they lack equivariance properties, representing a notable drawback in certain applications in chemistry and physical sciences. In this paper, we introduce the Clifford Group Equivariant Graph Neural Networks (CG-EGNNs), a novel EGNN that enhances high-order message passing by integrating high-order local structures in the context of Clifford algebras. As a key benefit of using Clifford algebras, CG-EGNN can learn functions that capture equivariance from positional features. By adopting the high-order message passing mechanism, CG-EGNN gains richer information from neighbors, thus improving model performance. Furthermore, we establish the universality property of the $k$-hop message passing framework, showcasing greater expressive power of CG-EGNNs with additional $k$-hop message passing mechanism. We empirically validate that CG-EGNNs outperform previous methods on various benchmarks including n-body, CMU motion capture, and MD17, highlighting their effectiveness in geometric deep learning.