  Partial differential equation (PDE) models and their associated variational
energy formulations are often rotationally invariant by design. This ensures
that a rotation of the input results in a corresponding rotation of the output,
which is desirable in applications such as image analysis. Convolutional neural
networks (CNNs) do not share this property, and existing remedies are often
complex. The goal of our paper is to investigate how diffusion and variational
models achieve rotation invariance and transfer these ideas to neural networks.
As a core novelty we propose activation functions which couple network channels
by combining information from several oriented filters. This guarantees
rotation invariance within the basic building blocks of the networks while
still allowing for directional filtering. The resulting neural architectures
are inherently rotationally invariant. With only a few small filters, they can
achieve the same invariance as existing techniques which require a fine-grained
sampling of orientations. Our findings help to translate diffusion and
variational models into mathematically well-founded network architectures, and
provide novel concepts for model-based CNN design.
