  We present a PDE-based framework that generalizes Group equivariant
Convolutional Neural Networks (G-CNNs). In this framework, a network layer is
seen as a set of PDE-solvers where geometrically meaningful PDE-coefficients
become the layer's trainable weights. Formulating our PDEs on homogeneous
spaces allows these networks to be designed with built-in symmetries such as
rotation in addition to the standard translation equivariance of CNNs.
  Having all the desired symmetries included in the design obviates the need to
include them by means of costly techniques such as data augmentation. We will
discuss our PDE-based G-CNNs (PDE-G-CNNs) in a general homogeneous space
setting while also going into the specifics of our primary case of interest:
roto-translation equivariance.
  We solve the PDE of interest by a combination of linear group convolutions
and non-linear morphological group convolutions with analytic kernel
approximations that we underpin with formal theorems. Our kernel approximations
allow for fast GPU-implementation of the PDE-solvers, we release our
implementation with this article in the form of the LieTorch extension to
PyTorch, available at https://gitlab.com/bsmetsjr/lietorch . Just like for
linear convolution a morphological convolution is specified by a kernel that we
train in our PDE-G-CNNs. In PDE-G-CNNs we do not use non-linearities such as
max/min-pooling and ReLUs as they are already subsumed by morphological
convolutions.
  We present a set of experiments to demonstrate the strength of the proposed
PDE-G-CNNs in increasing the performance of deep learning based imaging
applications with far fewer parameters than traditional CNNs.
