In this paper we design a neural interpolation operator to improve the boundary data for regional weather models, which is a challenging problem as we are required to map multi-scale dynamics between grid resolutions. In particular, we expose a methodology for approaching the problem through the study of a simplified model, with a view to generalise the results in this work to the dynamical core of regional weather models. Our approach will exploit a combination of techniques from image super-resolution with convolutional neural networks (CNNs) and residual networks, in addition to building the flow of atmospheric dynamics into the neural network