群等变性是一种强归纳偏置，在广泛的深度学习任务中非常有用。然而，为一般群和一般域构造高效的等变网络十分困难。Finzi等人（2021）的近期工作直接求解任意矩阵群的等变约束，从而得到等变MLP（EMLP）。但该方法扩展性不佳，而扩展性在深度学习中至关重要。本文提出群表示网络（G-RepsNet），这是一种面向任意矩阵群的轻量级等变网络，其特征使用张量多项式表示。我们设计的核心直觉是：在神经网络的隐藏层中使用张量表示，并结合简单且低成本张量运算，可以得到具有表达能力的通用等变网络。我们发现，在若干具有群对称性的任务上，例如以标量、向量和二阶张量作为数据类型的O(5)、O(1,3)和O(3)任务，G-RepsNet与EMLP具有竞争力。在图像分类任务上，我们发现使用二阶表示的G-RepsNet具有竞争力，并且常常优于复杂的最先进等变模型，例如GCNN（Cohen & Welling, 2016a）和E(2)-CNN（Weiler & Cesa, 2019）。为进一步说明我们方法的通用性，我们展示G-RepsNet在N体预测和求解偏微分方程上分别与G-FNO（Helwig et al., 2023）和EGNN（Satorras et al., 2021）具有竞争力，同时保持高效。
英文摘要：
Group equivariance is a strong inductive bias useful in a wide range of deep learning tasks. However, constructing efficient equivariant networks for general groups and domains is difficult. Recent work by Finzi et al. (2021) directly solves the equivariance constraint for arbitrary matrix groups to obtain equivariant MLPs (EMLPs). But this method does not scale well and scaling is crucial in deep learning. Here, we introduce Group Representation Networks (G-RepsNets), a lightweight equivariant network for arbitrary matrix groups with features represented using tensor polynomials. The key intuition for our design is that using tensor representations in the hidden layers of a neural network along with simple inexpensive tensor operations can lead to expressive universal equivariant networks. We find G-RepsNet to be competitive to EMLP on several tasks with group symmetries such as O(5), O(1, 3), and O(3) with scalars, vectors, and second-order tensors as data types. On image classification tasks, we find that G-RepsNet using second-order representations is competitive and often even outperforms sophisticated state-of-the-art equivariant models such as GCNNs (Cohen & Welling, 2016a) and E(2)-CNNs (Weiler & Cesa, 2019). To further illustrate the generality of our approach, we show that G-RepsNet is competitive to G-FNO (Helwig et al., 2023) and EGNN (Satorras et al., 2021) on N-body predictions and solving PDEs, respectively, while being efficient.
Facet: highly efficient E(3)-equivariant networks for interatomic potentials
Department of Computer Science and Engineering, University of South Carolina（南卡罗来纳大学计算机科学与工程系）；Computational Science and Engineering, Georgia Institute of Technology（佐治亚理工学院计算科学与工程系）
From Peptides to Nanostructures: A Euclidean Transformer for Fast and Stable Machine Learned Force Fields
TU Berlin（柏林工业大学）；BIFOLD, Berlin Institute for the Foundations of Learning and Data（柏林学习与数据基础研究所）；Google DeepMind（谷歌DeepMind）；Korea University（高丽大学）；Max Planck Institut für Informatik（马克斯·普朗克计算机科学研究所）
FlashCart: Fast Cartesian Tensor Products for Equivariant Interatomic Potentials
INM – Leibniz Institute for New Materials（莱布尼茨新材料研究所）；Saarland University（萨尔兰大学）；German Research Center for Artificial Intelligence (DFKI)（德国人工智能研究中心）；University of Stuttgart（斯图加特大学）；NEC Laboratories Europe（NEC欧洲实验室）
FPT Software AI Center（FPT软件人工智能中心）；McGill University（麦吉尔大学）；Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）；Ton Duc Thang University（同塔大学）；Indiana State University（印第安纳州立大学）
Multiset-Equivariant Set Prediction with Approximate Implicit Differentiation
Samsung - SAIT AI Lab（三星SAIT人工智能实验室）；Montreal University（蒙特利尔大学）；University of Amsterdam（阿姆斯特丹大学）；Mila（米拉人工智能研究所）；Université de Montreal（蒙特利尔大学）；TNO（荷兰应用科学研究院）
Frame-based Equivariant Diffusion Models for 3D Molecular Generation
University of Amsterdam（阿姆斯特丹大学）；AMLab；AI4Science Lab（AI4Science实验室）；Korteweg-de Vries Institute for Mathematics University of Amsterdam（Korteweg-de Vries数学研究所阿姆斯特丹大学）
ESGNN: Towards Equivariant Scene Graph Neural Network for 3D Scene Understanding
College of Engineering and Computer Science, VinUniversity（VinUniversity工程与计算机科学学院）；Department of Mathematics and Computer Science, Indiana State University（印第安纳州立大学数学与计算机科学系）
Equivariant Diffusion for Crystal Structure Prediction
School of Computer Science；Engineering, Sun Yat-sen University, Guangzhou, China；National Supercomputer Center in Guangzhou, China；Dept. of Comp. Sci. \& Tech., Institute for AI, Tsinghua University, Beijing, China；Institute for AIR, Tsinghua University, Beijing, China；Gaoling School of Artificial Intelligence, Renmin University of China, Beijing, China；Beijing Key Laboratory of Big Data Management；Analysis Methods, Beijing, China
University of Oxford（牛津大学）；ELLIS Institute（ELLIS研究所）；Max Planck Institute for Intelligent Systems（智能系统马克斯·普朗克研究所）；Tübingen AI Center（图宾根人工智能中心）；AITHYRA
The Price of Freedom: Exploring Expressivity and Runtime Tradeoffs in Equivariant Tensor Products
Department of EECS, Massachusetts Institute of Technology, Cambridge MA 02139, USA（麻省理工学院电子工程与计算机科学系）；Department of CSE, Massachusetts Institute of Technology, Cambridge MA 02139, USA（麻省理工学院计算机科学与工程系）
Enhancing lattice kinetic schemes for fluid dynamics with Lattice-Equivariant Neural Networks
Eindhoven University of Technology（埃因霍温理工大学）；SISSA (International School for Advanced Studies)（国际高等研究院（SISSA））；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）；Center for Nonlinear Studies (CNLS)（非线性研究中心（CNLS））；Eindhoven Artificial Intelligence Systems Institute（埃因霍温人工智能系统研究所）
Tensor Decomposition Networks for Fast Machine Learning Interatomic Potential Computations
Department of Computer Science and Engineering, Texas A&M University, USA（计算机科学与工程系，德克萨斯A&M大学）；Lambda, Inc., USA（Lambda公司）；RIKEN Cluster for Pioneering Research, RIKEN, Japan（日本RIKEN先锋研究中心）；NVIDIA, USA（NVIDIA公司）；Department of Materials Science and Engineering, Texas A&M University, USA（材料科学与工程系，德克萨斯A&M大学）；Mike Walker ’66 Department of Mechanical Engineering, Texas A&M University, USA（机械工程系，德克萨斯A&M大学）
A Coherence Law for Trainability in Noisy Equivariant Quantum Neural Networks
Centre for Visual Computing and Intelligent Systems（视觉计算与智能系统中心）；School of Individualized Study（个性化学习学院）；University of Bradford（布拉德福德大学）；Rochester Institute of Technology（罗切斯特理工学院）
TESGNN: Temporal Equivariant Scene Graph Neural Networks for Efficient and Robust Multi-View 3D Scene Understanding
Department of Robotics（机器人学系）；Mohamed bin Zayed University of Artificial Intelligence（莫扎德·本·扎伊德人工智能大学）；College of Engineering and Computer Science（工程与计算机科学学院）；VinUniversity（文大学）；Department of Computer Science（计算机科学系）；The University of Alabama at Birmingham（阿拉巴马大学伯明翰分校）
A Comparison Between Invariant and Equivariant Classical and Quantum Graph Neural Networks
University of Florida（佛罗里达大学）；Polytechnic University of Catalonia（加泰罗尼亚理工大学）；Indian Institute of Technology Bhilai（印度比莱印度理工学院）；University of Kansas（堪萨斯大学）；University of Alabama（阿拉巴马大学）；Software Engineering Institute, Carnegie Mellon University（卡内基梅隆大学软件工程研究所）
