Equivariant neural networks (ENNs) are graph neural networks embedded in $mathbb{R}^3$ and are well suited for predicting molecular properties. The ENN library e3nn has customizable convolutions, which can be designed to depend only on distances betw
een points, or also on angular features, making them rotationally invariant, or equivariant, respectively. This paper studies the practical value of including angular dependencies for molecular property prediction directly via an ablation study with texttt{e3nn} and the QM9 data set. We find that, for fixed network depth and parameter count, adding angular features decreased test error by an average of 23%. Meanwhile, increasing network depth decreased test error by only 4% on average, implying that rotationally equivariant layers are comparatively parameter efficient. We present an explanation of the accuracy improvement on the dipole moment, the target which benefited most from the introduction of angular features.
Curies principle states that when effects show certain asymmetry, this asymmetry must be found in the causes that gave rise to them. We demonstrate that symmetry equivariant neural networks uphold Curies principle and can be used to articulate many s
ymmetry-relevant scientific questions into simple optimization problems. We prove these properties mathematically and demonstrate them numerically by training a Euclidean symmetry equivariant neural network to learn symmetry-breaking input to deform a square into a rectangle and to generate octahedra tilting patterns in perovskites.
We present a general theory of Group equivariant Convolutional Neural Networks (G-CNNs) on homogeneous spaces such as Euclidean space and the sphere. Feature maps in these networks represent fields on a homogeneous base space, and layers are equivari
ant maps between spaces of fields. The theory enables a systematic classification of all existing G-CNNs in terms of their symmetry group, base space, and field type. We also consider a fundamental question: what is the most general kind of equivariant linear map between feature spaces (fields) of given types? Following Mackey, we show that such maps correspond one-to-one with convolutions using equivariant kernels, and characterize the space of such kernels.
We present a convolutional network that is equivariant to rigid body motions. The model uses scalar-, vector-, and tensor fields over 3D Euclidean space to represent data, and equivariant convolutions to map between such representations. These SE(3)-
equivariant convolutions utilize kernels which are parameterized as a linear combination of a complete steerable kernel basis, which is derived analytically in this paper. We prove that equivariant convolutions are the most general equivariant linear maps between fields over R^3. Our experimental results confirm the effectiveness of 3D Steerable CNNs for the problem of amino acid propensity prediction and protein structure classification, both of which have inherent SE(3) symmetry.
The success of convolutional networks in learning problems involving planar signals such as images is due to their ability to exploit the translation symmetry of the data distribution through weight sharing. Many areas of science and egineering deal
with signals with other symmetries, such as rotation invariant data on the sphere. Examples include climate and weather science, astrophysics, and chemistry. In this paper we present spherical convolutional networks. These networks use convolutions on the sphere and rotation group, which results in rotational weight sharing and rotation equivariance. Using a synthetic spherical MNIST dataset, we show that spherical convolutional networks are very effective at dealing with rotationally invariant classification problems.
