Improving regional weather forecasts with neural interpolation

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

Predictions Based on Pixel Data: Insights from PDEs and Finite Differences

Neural networks are the state-of-the-art for many approximation tasks in high-dimensional spaces, as supported by an abundance of experimental evidence. However, we still need a solid theoretical understanding of what they can approximate and, more importantly, at what cost and accuracy. One network architecture of practical use, especially for approximation tasks involving images, is convolutional (residual) networks. However, due to the locality of the linear operators involved in these networks, their analysis is more complicated than for generic fully connected neural networks. This paper focuses on sequence approximation tasks, where a matrix or a higher-order tensor represents each observation. We show that when approximating sequences arising from space-time discretisations of PDEs we may use relatively small networks. We constructively derive these results by exploiting connections between discrete convolution and finite difference operators. Throughout, we design our network architecture to, while having guarantees, be similar to those typically adopted in practice for sequence approximation tasks. Our theoretical results are supported by numerical experiments which simulate linear advection, the heat equation, and the Fisher equation. The implementation used is available at the repository associated to the paper.