  PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) utilize solvers of
geometrically meaningful evolution PDEs as substitutes for the conventional
components in G-CNNs. PDE-G-CNNs offer several key benefits all at once: fewer
parameters, inherent equivariance, better performance, data efficiency, and
geometric interpretability.
  In this article we focus on Euclidean equivariant PDE-G-CNNs where the
feature maps are two dimensional throughout. We call this variant of the
framework a PDE-CNN.
  From a machine learning perspective, we list several practically desirable
axioms and derive from these which PDEs should be used in a PDE-CNN. Here our
approach to geometric learning via PDEs is inspired by the axioms of classical
linear and morphological scale-space theory, which we generalize by introducing
semifield-valued signals.
  Furthermore, we experimentally confirm for small networks that PDE-CNNs offer
fewer parameters, increased performance, and better data efficiency when
compared to CNNs. We also investigate what effect the use of different
semifields has on the performance of the models.
