We consider PDE-based Group Convolutional Neural Networks (PDE-G-CNNs) that generalize Group equivariant Convolutional Neural Networks (G-CNNs). In PDE-G-CNNs a network layer is a set of PDE-solvers.
The underlying (non)linear PDEs are defined on the homogeneous space M(d) of positions and orientations within the roto-translation group SE(d) and provide a geometric design of the roto-translation equivariant neural network.
The network consists of morphological convolutions with (approximative) kernels solving nonlinear PDEs (HJB equations for max-pooling over Riemannian balls), and linear convolutions solving linear PDEs (convection, fractional diffusion). Our analytic approximation kernels are accurate in comparison to our recent exact PDE-kernels. Common mystifying (ReLU) nonlinearities are now obsolete and excluded. We achieve high data-efficiency of our networks: better classification results in image processing with both less training data and less network complexity. Moreover, we have network interpretability as we train sparse association fields (modeling contour perception in our own visual system).