We investigate numerous structural connections between numerical algorithms for partial differential equations (PDEs) and
neural architectures. Our goal is to transfer the rich set of mathematical foundations from the world of PDEs to neural networks.
Besides structural insights, we provide concrete examples and experimental evaluations of the resulting architectures. Using
the example of generalised nonlinear diffusion in 1D, we consider explicit schemes, acceleration strategies thereof, implicit
schemes, and multigrid approaches. We connect these concepts to residual networks, recurrent neural networks, and U-net
architectures. Our findings inspire a symmetric residual network design with provable stability guarantees and justify the
effectiveness of skip connections in neural networks from a numerical perspective. Moreover, we present U-net architectures
that implement multigrid techniques for learning efficient solutions of partial differential equation models, and motivate
uncommon design choices such as trainable nonmonotone activation functions. Experimental evaluations show that the
proposed architectures save half of the trainable parameters and can thus outperform standard ones with the same model
complexity. Our considerations serve as a basis for explaining the success of popular neural architectures and provide a
blueprint for developing new mathematically well-founded neural building blocks