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 network interpretability as we train sparse association fields (modeling contour perception in our own visual system). We present several medical imaging applications (tracking and segmentation of complex vasculature) and industrial image analysis applications where each time we show benefits of PDE-G-CNNs compared to state-of-the-art G-CNNs: increase of performance along with a large reduction in network parameters and less training data. Associated publications in my group: [1] PDE-Based Group Equivariant Convolutional Networks (PDE-G-CNNs) (2023) [2] Analysis of (sub-)Riemannian PDE-G-CNNs (2023) [3] Functional Properties of PDE-Based Group Equivariant Convolutional Neural Networks (2023) [3] Geodesic Tracking via New Data-driven Connections of Cartan Type for Vascular Tree Tracking (2023) [4] New exact and numerical solutions of the (convection–) diffusion kernels on SE(3) (2019) [5] Equivariant Deep Learning via Morphological and Linear Scale Space PDEs on M(d) (2019) [6] Roto-translation covariant convolutional networks for medical image analysis (G-CNNs) (2018)