  Spherical signals exist in many applications, e.g., planetary data, LiDAR
scans and digitalization of 3D objects, calling for models that can process
spherical data effectively. It does not perform well when simply projecting
spherical data into the 2D plane and then using planar convolution neural
networks (CNNs), because of the distortion from projection and ineffective
translation equivariance. Actually, good principles of designing spherical CNNs
are avoiding distortions and converting the shift equivariance property in
planar CNNs to rotation equivariance in the spherical domain. In this work, we
use partial differential operators (PDOs) to design a spherical equivariant
CNN, PDO-e$\text{S}^\text{2}$CNN, which is exactly rotation equivariant in the
continuous domain. We then discretize PDO-e$\text{S}^\text{2}$CNNs, and analyze
the equivariance error resulted from discretization. This is the first time
that the equivariance error is theoretically analyzed in the spherical domain.
In experiments, PDO-e$\text{S}^\text{2}$CNNs show greater parameter efficiency
and outperform other spherical CNNs significantly on several tasks.
