It is increasingly recognised that there is a close relationship between some network architectures and iterative solvers for partial differential equations. In this talk we present a network architecture for forward and inverse problems in non-linear diffusion. By design the architecture is non-linear, learning an anisotropic diffusivity function for each layer from the output of the previous layer. The performed updates are explicit, by which we obtain better interpretability and generalisability compared to classical architectures. Since backward diffusion is unstable, a learned regularisation is implicitly learned to stabilise this process. We test results on synthetic image data sets that have undergone edge-preserving diffusion and on experimental data of images view through variable density scattering media.