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 fall 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 E(3)-equivariance with LLM capabilities. Specifically, EquiLLM comprises four key components: geometry-aware prompting, an equivariant encoder, an LLM, and an equivariant adaptor. 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 adaptor 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) 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.

Geometric graph is a special kind of graph with geometric features, which is 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 tackle this issue, researchers proposed a variety of Geometric Graph Neural Networks 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 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.