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 Fluid113 K benchmark (113,000 nodes). Results demonstrate superior efficiency and performance, establishing new capabilities in large-scale equivariant graph learning.

Understanding the 3D structure of RNA is essential for deciphering its function and developing RNA-based therapeutics. Geometric Graph Neural Networks (GeoGNNs) that conform to the $\mathrm{E}(3)$-symmetry have advanced RNA structure evaluation, a crucial step toward RNA structure prediction. However, existing GeoGNNs are still defective in two aspects: 1. inefficient or incapable of capturing the full geometries of RNA; 2. limited generalization ability when the size of RNA significantly differs between training and test datasets. In this paper, we propose EquiRNA, a novel equivariant GNN model by exploring the three-level hierarchical geometries of RNA. At its core, EquiRNA effectively addresses the size generalization challenge by reusing the representation of nucleotide, the common building block shared across RNAs of varying sizes. Moreover, by adopting a scalarization-based equivariant GNN as the backbone, our model maintains directional information while offering higher computational efficiency compared to existing GeoGNNs. Additionally, we propose a size-insensitive $K$-nearest neighbor sampling strategy to enhance the model's robustness to RNA size shifts. We test our approach on our created benchmark as well as an existing dataset. The results show that our method significantly outperforms other state-of-the-art methods, providing a robust baseline for RNA 3D structure modeling and evaluation.

Accurately predicting 3D structures and dynamics of physical systems is crucial in scientific applications. Existing approaches that rely on geometric Graph Neural Networks (GNNs) effectively enforce $\mathrm{E}(3)$-equivariance, but they often fail in leveraging extensive broader information. While direct application of Large Language Models (LLMs) can incorporate external knowledge, they lack the capability for spatial reasoning with guaranteed equivariance. In this paper, we propose EquiLLM, a novel framework for representing 3D physical systems that seamlessly integrates $\mathrm{E}(3)$-equivariance with LLM capabilities. Specifically, EquiLLM comprises four key components: geometry-aware prompting, an equivariant encoder, an LLM, and an equivariant adapter. Essentially, the LLM guided by the instructive prompt serves as a sophisticated invariant feature processor, while 3D directional information is exclusively handled by the equivariant encoder and adapter modules. Experimental results demonstrate that EquiLLM delivers significant improvements over previous methods across molecular dynamics simulation, human motion simulation, and antibody design, highlighting its promising generalizability.

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.

Accurate prediction of ionic conductivity in electrolyte systems is crucial for advancing numerous scientific and technological applications. While significant progress has been made, current research faces two fundamental challenges: (1) the lack of high-quality standardized benchmarks, and (2) inadequate modeling of geometric structure and intermolecular interactions in mixture systems. To address these limitations, we first reorganize and enhance the CALiSol and DiffMix electrolyte datasets by incorporating geometric graph representations of molecules. We then propose GeoMix, a novel geometry-aware framework that preserves Set-SE(3) equivariance—an essential but challenging property for mixture systems. At the heart of GeoMix lies the Geometric Interaction Network (GIN), an equivariant module specifically designed for intermolecular geometric message passing. Comprehensive experiments demonstrate that GeoMix consistently outperforms diverse baselines (including MLPs, GNNs, and geometric GNNs) across both datasets, validating the importance of cross-molecular geometric interactions and equivariant message passing for accurate property prediction. This work not only establishes new benchmarks for electrolyte research but also provides a general geometric learning framework that advances modeling of mixture systems in energy materials, pharmaceutical development, and beyond.

Abstract Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections, making them ineffectively processed by current Graph Neural Networks (GNNs). To address this issue, researchers proposed a variety of geometric GNNs equipped with invariant/equivariant properties to better characterize the geometry and topology of geometric graphs. Given the current progress in this field, it is imperative to conduct a comprehensive survey of data structures, models, and applications related to geometric GNNs. In this paper, based on the necessary but concise mathematical preliminaries, we formalize geometric graph as the data structure, on top of which we provide a unified view of existing models from the geometric message passing perspective. Additionally, we summarize the applications as well as the related datasets to facilitate later research for methodology development and experimental evaluation. We also discuss the challenges and future potential directions of geometric GNNs at the end of this survey.

Equivariant Graph Neural Networks (GNNs) have made remarkable success in a variety of scientific applications. However, existing equivariant GNNs encounter the efficiency issue for large geometric graphs and perform poorly if the input is reduced to sparse local graph for speed acceleration. In this paper, we propose FastEGNN, an enhanced model of equivariant GNNs on large geometric graphs. The central idea is leveraging a small ordered set of virtual nodes to approximate the large unordered graph of real nodes. In particular, we distinguish the message passing and aggregation for different virtual node to encourage the mutual distinctiveness, and minimize the Maximum Mean Discrepancy (MMD) between virtual and real coordinates to realize the global distributedness. FastEGNN meets all necessary E(3) symmetries, with certain universal expressivity assurance as well. Our experiments on N-body systems (100 nodes), proteins (800 nodes) and water-3D (8000 nodes), demonstrate that FastEGNN achieves a promising balance between accuracy and efficiency, and outperforms EGNN in accuracy even after dropping all edges in real systems like proteins and water-3D.

Equivariant Graph Neural Networks (GNNs) that incorporate E(3) symmetry have achieved significant success in various scientific applications. As one of the most successful models, EGNN leverages a simple scalarization technique to perform equivariant message passing over only Cartesian vectors (i.e., 1st-degree steerable vectors), enjoying greater efficiency and efficacy compared to equivariant GNNs using higher-degree steerable vectors. This success suggests that higher-degree representations might be unnecessary. In this paper, we disprove this hypothesis by exploring the expressivity of equivariant GNNs on symmetric structures, including $k$-fold rotations and regular polyhedra. We theoretically demonstrate that equivariant GNNs will always degenerate to a zero function if the degree of the output representations is fixed to 1 or other specific values. Based on this theoretical insight, we propose HEGNN, a high-degree version of EGNN to increase the expressivity by incorporating high-degree steerable vectors while maintaining EGNN's efficiency through the scalarization trick. Our extensive experiments demonstrate that HEGNN not only aligns with our theoretical analyses on toy datasets consisting of symmetric structures, but also shows substantial improvements on more complicated datasets such as $N$-body and MD17. Our theoretical findings and empirical results potentially open up new possibilities for the research of equivariant GNNs.