[论文解读] E(n) Equivariant Graph Neural Networks

Victor García Satorras, Emiel Hoogeboom+1 more|arXiv (Cornell University)|Feb 19, 2021

Machine Learning in Materials Science参考文献 37被引用 105

一句话总结

EGNN 引入了一种 E(n)-等变的图神经网络，它以更新坐标和特征的方式，同时保持平移、旋转、反射和置换等变换的等变性，且不依赖球谐函数，并可扩展到更高维度。

ABSTRACT

This paper introduces a new model to learn graph neural networks equivariant to rotations, translations, reflections and permutations called E(n)-Equivariant Graph Neural Networks (EGNNs). In contrast with existing methods, our work does not require computationally expensive higher-order representations in intermediate layers while it still achieves competitive or better performance. In addition, whereas existing methods are limited to equivariance on 3 dimensional spaces, our model is easily scaled to higher-dimensional spaces. We demonstrate the effectiveness of our method on dynamical systems modelling, representation learning in graph autoencoders and predicting molecular properties.

研究动机与目标

• Motivate and design a graph neural network that is equivariant to E(n) transformations (rotations, translations, reflections) and permutations.
• Avoid expensive higher-order representations while maintaining competitive performance.
• Demonstrate effectiveness across dynamical systems, graph autoencoders, and molecular property prediction in QM9.
• Show scalability to spaces with dimensionality n > 3 and compare against existing equivariant methods.

提出的方法

• Introduce an Equivariant Graph Convolutional Layer (EGCL) that processes node features h^l, coordinates x^l, and edge information E.
• Edge updates use m_{ij} = φ_e(h_i^l, h_j^l, ||x_i^l - x_j^l||^2, a_{ij}).
• Coordinate updates are x_i^{l+1} = x_i^l + C sum_{j≠i} (x_i^l - x_j^l) φ_x(m_{ij}) with C = 1/(M-1).
• Aggregate messages to form m_i = sum_j m_{ij} and update node features h_i^{l+1} = φ_h(h_i^l, m_i).
• Optionally extend with particle momentum: v_i^{l+1} = φ_v(h_i^l) v_i^{init} +… and x_i^{l+1} = x_i^l + v_i^{l+1}.
• Include a mechanism to infer edges when adjacency is not provided via φ_inf to produce soft edge values.]
• research_questions:[

实验结果

研究问题

• RQ1 Can EGNNs achieve E(n) (including SE(3) with reflections) equivariance without spherical harmonics or high-order representations?
• RQ2 Do EGNNs maintain or improve performance relative to non-equivariant GNNs and existing E(3)-equivariant models across dynamic, autoencoding, and molecular tasks?
• RQ3 Is the method scalable to higher dimensional spaces beyond 3D while remaining computationally efficient?
• RQ4 How does integrating coordinates and feature messages within the edge and node updates affect data efficiency and predictive accuracy?
• RQ5 Can edges be inferred on-the-fly without explicit graphs and still preserve equivariance?]
• RQ6 key_findings:["EGNN achieves competitive or superior performance compared to both non-equivariant GNNs and some equivariant baselines across dynamical N-body tasks, graph autoencoding, and QM9 molecular prediction.","On a 3D charged-particle N-body dataset, EGNN attains the lowest MSE (0.0071) among listed methods and is faster than several high-cost equivariant models.","The model remains E(n) equivariant with respect to translations, rotations, and reflections, and scalable to higher dimensions without relying on spherical harmonics.","In a graph autoencoder setup, EGNN substantially improves edge-reconstruction metrics over GNN variants, and handles symmetry-breaking noise in a translation/rotation-equivariant manner.","On the QM9 benchmark, EGNN achieves competitive mean absolute errors for molecular properties, outperforming several baselines in key metrics (e.g., MAE ≈ 0.071 for α).","EGNN supports an edge-inference mechanism to learn graph connectivity when adjacency is not provided, while preserving E(n) equivariance."]
• RQ7 table_headers: []
• RQ8 table_rows: []}
• RQ9

主要发现

• EGNN 在动力学N体任务、图自编码和QM9分子预测等方面，与非等变GNNs和部分等变基线相比，表现具有竞争力或更优。
• 在3D带电粒子N体数据集上，EGNN 达到列出方法中的最低MSE（0.0071），且比若干高成本的等变模型更快。
• 该模型在平移、旋转和反射方面保持E(n)等变，并且可扩展到更高维度，而无需依赖球谐函数。
• 在图自编码器设置中，EGNN 在边重构指标方面显著优于GNN变体，并以可平移/旋转等变的方式处理对称性破缺噪声。
• 在QM9基准上，EGNN 在分子性质的平均绝对误差方面具有竞争力，优于若干基线的关键指标（如 α 的 MAE 约为 0.071）。
• EGNN 支持边缘推断机制，在未给出邻接信息时也能学习图连通性，同时保持 E(n) 等变性。