Convolutional Neural Networks (CNNs) filter the input data using a series of spatial convolution operators with compactly supported stencils and point-wise nonlinearities. Commonly, the convolution operators couple features from all channels. For wide networks, this leads to immense computational cost in the training of and prediction with CNNs. In this paper, we present novel ways to parameterize the convolution more efficiently, aiming to decrease the number of parameters in CNNs and their computational complexity. We propose new architectures that use a sparser coupling between the channels and thereby reduce both the number of trainable weights and the computational cost of the CNN. Our architectures arise as new types of residual neural network (ResNet) that can be seen as discretizations of a Partial Differential Equations (PDEs) and thus have predictable theoretical properties. Our first architecture involves a convolution operator with a special sparsity structure, and is applicable to a large class of CNNs. Next, we present an architecture that can be seen as a discretization of a diffusion reaction PDE, and use it with three different convolution operators. We outline in our experiments that the proposed architectures, although considerably reducing the number of trainable weights, yield comparable accuracy to existing CNNs that are fully coupled in the channel dimension.