The prediction of quantum mechanical properties is historically plagued by a trade-off between accuracy and speed. Machine learning potentials have previously shown great success in this domain, reaching increasingly better accuracy while maintaining computational efficiency comparable with classical force fields. In this work we propose TorchMD-NET, a novel equivariant transformer (ET) architecture, outperforming state-of-the-art on MD17, ANI-1, and many QM9 targets in both accuracy and computational efficiency. Through an extensive attention weight analysis, we gain valuable insights into the black box predictor and show differences in the learned representation of conformers versus conformations sampled from molecular dynamics or normal modes. Furthermore, we highlight the importance of datasets including off-equilibrium conformations for the evaluation of molecular potentials.
Universitat Pompeu Fabra（庞培法布拉大学）；Freie Universität（柏林自由大学）；National Research Council (CNR-IBF)（国家研究委员会（CNR-IBF））
Molecular dynamics simulations provide a mechanistic description of molecules by relying on empirical potentials. The quality and transferability of such potentials can be improved leveraging data-driven models derived with machine learning approaches. Here, we present TorchMD, a framework for molecular simulations with mixed classical and machine learning potentials. All of force computations including bond, angle, dihedral, Lennard-Jones and Coulomb interactions are expressed as PyTorch arrays and operations. Moreover, TorchMD enables learning and simulating neural network potentials. We val…
Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State University（罗蒙诺索夫莫斯科国立大学斯科贝尔琴核物理研究所）
Currently there exists rather promising new trend in machine leaning (ML) based on the relationship between neural networks (NN) and Gaussian processes (GP), including many related subtopics, e.g., signal propagation in NNs, theoretical derivation of learning curve for NNs, QFT methods in ML, etc. An important feature of convolutional neural networks (CNN) is their equivariance (consistency) with respect to the symmetry transformations of the input data. In this work we establish a relationship between the many-channel limit for CNNs equivariant with respect to two-dimensional Euclidean group …
Convolutional neural networks revolutionized computer vision and natrual language processing. Their efficiency, as compared to fully connected neural networks, has its origin in the architecture, where convolutions reflect the translation invariance in space and time in pattern or speech recognition tasks. Recently, Cohen and Welling have put this in the broader perspective of invariance under symmetry groups, which leads to the concept of group equivaiant neural networks and more generally steerable neural networks. In this article, we review the architecture of such networks including equiva…
Equivariant Spatio-Temporal Attentive Graph Networks to Simulate Physical Dynamics
Gaoling School of Artificial Intelligence, Renmin University of China（中国人民大学高瓴人工智能学院）；North Carolina State University（北卡罗来纳州立大学）；Institute for AI Industry Research (AIR), Tsinghua University（清华大学人工智能产业研究院）；Tencent AI Lab（腾讯AI实验室）
Stanford University（斯坦福大学）；University of California, Berkeley（加州大学伯克利分校）；Lawrence Berkeley National Lab（劳伦斯伯克利国家实验室）；Google（谷歌）
We introduce tensor field neural networks, which are locally equivariant to 3D rotations, translations, and permutations of points at every layer. 3D rotation equivariance removes the need for data augmentation to identify features in arbitrary orientations. Our network uses filters built from spherical harmonics; due to the mathematical consequences of this filter choice, each layer accepts as input (and guarantees as output) scalars, vectors, and higher-order tensors, in the geometric sense of these terms. We demonstrate the capabilities of tensor field networks with tasks in geometry, physi…
McGill University（麦吉尔大学）；Mila Quebec AI Institute（米拉魁北克人工智能研究所）；RWTH Aachen University（亚琛工业大学）；Max Planck Institute for Software Systems（马克斯·普朗克软件系统研究所）
While (message-passing) graph neural networks have clear limitations in approximating permutation-equivariant functions over graphs or general relational data, more expressive, higher-order graph neural networks do not scale to large graphs. They either operate on $k$-order tensors or consider all $k$-node subgraphs, implying an exponential dependence on $k$ in memory requirements, and do not adapt to the sparsity of the graph. By introducing new heuristics for the graph isomorphism problem, we devise a class of universal, permutation-equivariant graph networks, which, unlike previous architec…
Chinese Academy of Sciences（中国科学院）；Xi’an Jiaotong University（西安交通大学）；George Mason University（乔治梅森大学）；Microsoft Research（微软研究院）
Group equivariance (e.g. SE(3) equivariance) is a critical physical symmetry in science, from classical and quantum physics to computational biology. It enables robust and accurate prediction under arbitrary reference transformations. In light of this, great efforts have been put on encoding this symmetry into deep neural networks, which has been shown to improve the generalization performance and data efficiency for downstream tasks. Constructing an equivariant neural network generally brings high computational costs to ensure expressiveness. Therefore, how to better trade-off the expressiven…
This work seeks to improve the generalization and robustness of existing neural networks for 3D point clouds by inducing group equivariance under general group transformations. The main challenge when designing equivariant models for point clouds is how to trade-off the performance of the model and the complexity. Existing equivariant models are either too complicate to implement or very high complexity. The main aim of this study is to build a general procedure to introduce group equivariant property to SOTA models for 3D point clouds. The group equivariant models built form our procedure are…
GLGENN: A Novel Parameter-Light Equivariant Neural Networks Architecture Based on Clifford Geometric Algebras
HSE University（莫斯科高等经济大学）；Skolkovo Institute of Science（斯克尔科沃科学研究院）；Institute for Information Transmission Problems of the Russian Academy of Sciences（俄罗斯科学院信息传输问题研究所）
RIKEN Center for Advanced Intelligence Project（理化学研究所先进智能项目中心）；Keio University（庆应义塾大学）；École Normale Supérieure Paris-Saclay（巴黎萨克雷高等师范学院）
In this paper, we develop a theory about the relationship between $G$-invariant/equivariant functions and deep neural networks for finite group $G$. Especially, for a given $G$-invariant/equivariant function, we construct its universal approximator by deep neural network whose layers equip $G$-actions and each affine transformations are $G$-equivariant/invariant. Due to representation theory, we can show that this approximator has exponentially fewer free parameters than usual models.
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Purdue University（普渡大学）
We propose learnable polyphase sampling (LPS), a pair of learnable down/upsampling layers that enable truly shift-invariant and equivariant convolutional networks. LPS can be trained end-to-end from data and generalizes existing handcrafted downsampling layers. It is widely applicable as it can be integrated into any convolutional network by replacing down/upsampling layers. We evaluate LPS on image classification and semantic segmentation. Experiments show that LPS is on-par with or outperforms existing methods in both performance and shift consistency. For the first time, we achieve true shi…
Center for Mathematical Morphology（数学形态学中心）；Mines ParisTech（巴黎高等矿业学院）；PSL University（巴黎文理研究大学）
In neural networks, the property of being equivariant to transformations improves generalization when the corresponding symmetry is present in the data. In particular, scale-equivariant networks are suited to computer vision tasks where the same classes of objects appear at different scales, like in most semantic segmentation tasks. Recently, convolutional layers equivariant to a semigroup of scalings and translations have been proposed. However, the equivariance of subsampling and upsampling has never been explicitly studied even though they are necessary building blocks in some segmentation …
One of the most discussed issues in graph generative modeling is the ordering of the representation. One solution consists of using equivariant generative functions, which ensure the ordering invariance. After having discussed some properties of such functions, we propose 3G-GAN, a 3-stages model relying on GANs and equivariant functions. The model is still under development. However, we present some encouraging exploratory experiments and discuss the issues still to be addressed.
Tsinghua University（清华大学）；Renmin University of China（中国人民大学）；Tencent AI Lab（腾讯人工智能实验室）
Equivariant Graph neural Networks (EGNs) are powerful in characterizing the dynamics of multi-body physical systems. Existing EGNs conduct flat message passing, which, yet, is unable to capture the spatial/dynamical hierarchy for complex systems particularly, limiting substructure discovery and global information fusion. In this paper, we propose Equivariant Hierarchy-based Graph Networks (EGHNs) which consist of the three key components: generalized Equivariant Matrix Message Passing (EMMP) , E-Pool and E-UpPool. In particular, EMMP is able to improve the expressivity of conventional equivari…
University of California San Diego（加州大学圣地亚哥分校）；Northeastern University（东北大学）
Exploiting symmetry in dynamical systems is a powerful way to improve the generalization of deep learning. The model learns to be invariant to transformation and hence is more robust to distribution shift. Data augmentation and equivariant networks are two major approaches to injecting symmetry into learning. However, their exact role in improving generalization is not well understood. In this work, we derive the generalization bounds for data augmentation and equivariant networks, characterizing their effect on learning in a unified framework. Unlike most prior theories for the i.i.d. setting…
Chalmers University of Technology（查尔姆斯理工大学）；Gothenburg University（哥德堡大学）；Umeå University（于默奥大学）；Zenseact
We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds $\mathcal{M}$ using principal bundles with structure group $K$ and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces $\mathcal{M}=G/K$, which are instead equivariant with respect to the global symmetry $G$ on $\mathcal{M}$. Group equivariant layers can be interpreted as intertwiners between ind…
École Normale Supérieure, PSL（巴黎高等师范学院（PSL大学））；Mila（米拉研究所）；Université de Montréal（蒙特利尔大学）；McGill University（麦吉尔大学）
The Strong Lottery Ticket Hypothesis (SLTH) stipulates the existence of a subnetwork within a sufficiently overparameterized (dense) neural network that -- when initialized randomly and without any training -- achieves the accuracy of a fully trained target network. Recent works by Da Cunha et. al 2022; Burkholz 2022 demonstrate that the SLTH can be extended to translation equivariant networks -- i.e. CNNs -- with the same level of overparametrization as needed for the SLTs in dense networks. However, modern neural networks are capable of incorporating more than just translation symmetry, and …
Khoury College of Computer Sciences, Northeastern University（东北大学库利计算机学院）；Informatics Institute, University of Amsterdam（阿姆斯特丹大学信息学研究所）
Incorporating symmetries can lead to highly data-efficient and generalizable models by defining equivalence classes of data samples related by transformations. However, characterizing how transformations act on input data is often difficult, limiting the applicability of equivariant models. We propose learning symmetric embedding networks (SENs) that encode an input space (e.g. images), where we do not know the effect of transformations (e.g. rotations), to a feature space that transforms in a known manner under these operations. This network can be trained end-to-end with an equivariant task …
Technion – Israel Institute of Technology（以色列理工学院）；NVIDIA Research（英伟达研究院）
Equivariance to permutations and rigid motions is an important inductive bias for various 3D learning problems. Recently it has been shown that the equivariant Tensor Field Network architecture is universal -- it can approximate any equivariant function. In this paper we suggest a much simpler architecture, prove that it enjoys the same universality guarantees and evaluate its performance on Modelnet40. The code to reproduce our experiments is available at \url{https://github.com/simpleinvariance/UniversalNetwork}
Michigan State University（密歇根州立大学）；Meta AI；Ford Motor Company（福特汽车公司）
Modern neural networks use building blocks such as convolutions that are equivariant to arbitrary 2D translations. However, these vanilla blocks are not equivariant to arbitrary 3D translations in the projective manifold. Even then, all monocular 3D detectors use vanilla blocks to obtain the 3D coordinates, a task for which the vanilla blocks are not designed for. This paper takes the first step towards convolutions equivariant to arbitrary 3D translations in the projective manifold. Since the depth is the hardest to estimate for monocular detection, this paper proposes Depth EquiVarIAnt NeTwo…
McGill University（麦吉尔大学）；Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）
Pixelizations of Platonic solids such as the cube and icosahedron have been widely used to represent spherical data, from climate records to Cosmic Microwave Background maps. Platonic solids have well-known global symmetries. Once we pixelize each face of the solid, each face also possesses its own local symmetries in the form of Euclidean isometries. One way to combine these symmetries is through a hierarchy. However, this approach does not adequately model the interplay between the two levels of symmetry transformations. We show how to model this interplay using ideas from group theory, iden…
Tsinghua University（清华大学）；Tencent AI Lab（腾讯人工智能实验室）
Many scientific problems require to process data in the form of geometric graphs. Unlike generic graph data, geometric graphs exhibit symmetries of translations, rotations, and/or reflections. Researchers have leveraged such inductive bias and developed geometrically equivariant Graph Neural Networks (GNNs) to better characterize the geometry and topology of geometric graphs. Despite fruitful achievements, it still lacks a survey to depict how equivariant GNNs are progressed, which in turn hinders the further development of equivariant GNNs. To this end, based on the necessary but concise math…
University of California San Diego（加州大学圣地亚哥分校）；Northeastern University（东北大学）
Incorporating symmetry as an inductive bias into neural network architecture has led to improvements in generalization, data efficiency, and physical consistency in dynamics modeling. Methods such as CNNs or equivariant neural networks use weight tying to enforce symmetries such as shift invariance or rotational equivariance. However, despite the fact that physical laws obey many symmetries, real-world dynamical data rarely conforms to strict mathematical symmetry either due to noisy or incomplete data or to symmetry breaking features in the underlying dynamical system. We explore approximatel…
Chalmers University of Technology（查尔姆斯理工大学）；TU Wien（维也纳工业大学）
Lattice gauge equivariant convolutional neural networks (L-CNNs) are a framework for convolutional neural networks that can be applied to non-Abelian lattice gauge theories without violating gauge symmetry. We demonstrate how L-CNNs can be equipped with global group equivariance. This allows us to extend the formulation to be equivariant not just under translations but under global lattice symmetries such as rotations and reflections. Additionally, we provide a geometric formulation of L-CNNs and show how convolutions in L-CNNs arise as a special case of gauge equivariant neural networks on SU…
Equivariant networks capture the inductive bias about the symmetry of the learning task by building those symmetries into the model. In this paper, we study how equivariance relates to generalization error utilizing PAC Bayesian analysis for equivariant networks, where the transformation laws of feature spaces are determined by group representations. By using perturbation analysis of equivariant networks in Fourier domain for each layer, we derive norm-based PAC-Bayesian generalization bounds. The bound characterizes the impact of group size, and multiplicity and degree of irreducible represen…
Weizmann Institute of Science（魏茨曼科学研究所）；Department of Computer Science and Applied Mathematics（计算机科学与应用数学系）
Using deep neural networks that are either invariant or equivariant to permutations in order to learn functions on unordered sets has become prevalent. The most popular, basic models are DeepSets [Zaheer et al. 2017] and PointNet [Qi et al. 2017]. While known to be universal for approximating invariant functions, DeepSets and PointNet are not known to be universal when approximating \emph{equivariant} set functions. On the other hand, several recent equivariant set architectures have been proven equivariant universal [Sannai et al. 2019], [Keriven et al. 2019], however these models either use …
UvA-Bosch Delta Lab（阿姆斯特丹大学-博世德尔塔实验室）；University of Amsterdam（阿姆斯特丹大学）
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 syste…
RIKEN AIP（理化学研究所人工智能促进项目）；The University of Tokyo（东京大学）
Invariant and equivariant networks are useful in learning data with symmetry, including images, sets, point clouds, and graphs. In this paper, we consider invariant and equivariant networks for symmetries of finite groups. Invariant and equivariant networks have been constructed by various researchers using Reynolds operators. However, Reynolds operators are computationally expensive when the order of the group is large because they use the sum over the whole group, which poses an implementation difficulty. To overcome this difficulty, we consider representing the Reynolds operator as a sum ov…
We prove an impossibility result, which in the context of function learning says the following: under certain conditions, it is impossible to simultaneously learn symmetries and functions equivariant under them using an ansatz consisting of equivariant functions. To formalize this statement, we carefully study notions of approximation for groups and semigroups. We analyze certain families of neural networks for whether they satisfy the conditions of the impossibility result: what we call ``linearly equivariant'' networks, and group-convolutional networks. A lot can be said precisely about line…
United Kingdom Atomic Energy Authority（英国原子能管理局）；The University of Hong Kong（香港大学）
We devise a new type of feedforward neural network. It is equivariant with respect to the unitary group $U(n)$. The input and output can be vectors in $\mathbb{C}^n$ with arbitrary dimension $n$. No convolution layer is required in our implementation. We avoid errors due to truncated higher order terms in Fourier-like transformation. The implementation of each layer can be done efficiently using simple calculations. As a proof of concept, we have given empirical results on the prediction of the dynamics of atomic motion to demonstrate the practicality of our approach.
Invariant and equivariant networks have been successfully used for learning images, sets, point clouds, and graphs. A basic challenge in developing such networks is finding the maximal collection of invariant and equivariant linear layers. Although this question is answered for the first three examples (for popular transformations, at-least), a full characterization of invariant and equivariant linear layers for graphs is not known. In this paper we provide a characterization of all permutation invariant and equivariant linear layers for (hyper-)graph data, and show that their dimension, in ca…
Many problems in computer vision and machine learning can be cast as learning on hypergraphs that represent higher-order relations. Recent approaches for hypergraph learning extend graph neural networks based on message passing, which is simple yet fundamentally limited in modeling long-range dependencies and expressive power. On the other hand, tensor-based equivariant neural networks enjoy maximal expressiveness, but their application has been limited in hypergraphs due to heavy computation and strict assumptions on fixed-order hyperedges. We resolve these problems and present Equivariant Hy…
In this work we develop a new method, named Sub-graph Permutation Equivariant Networks (SPEN), which provides a framework for building graph neural networks that operate on sub-graphs, while using a base update function that is permutation equivariant, that are equivariant to a novel choice of automorphism group. Message passing neural networks have been shown to be limited in their expressive power and recent approaches to over come this either lack scalability or require structural information to be encoded into the feature space. The general framework presented here overcomes the scalabilit…
