SO(3)-Equivariant Neural Networks for Learning from Scalar and Vector Fields on Spheres

Abstract

Analyzing scalar and vector fields on the sphere, such as temperature or wind speed and direction on Earth, is a difficult task. Models should respect both the rotational symmetries of the sphere and the inherent symmetries of the vector fields. A class of equivariant models has emerged, which process these spherical signals by applying group convolutions in Fourier space with respect to the three-dimensional rotation group. However, the proposed models are constrained in the choice of convolution kernels and nonlinearities in order to preserve the desired signal properties. In this paper, we introduce a deep learning architecture without these limitations, thus with a richer class of convolution kernels and activation functions. This architecture is suitable for signals consisting of both scalar and vector fields on the sphere, as they can be described as equivariant signals on the three-dimensional rotation group. Experiments show that this architecture generally outperforms standard CNNs and often matches or exceeds the performance of spherical CNNs trained under comparable conditions. However, the advantage over sCNNs is not uniform across all tasks and we observe that incorporating the interaction between different spins in the hidden layers narrows this gap.