James Jackaman

James Jackaman, Georgios Papamikos, Tristan Pryer

We design a consistent Galerkin scheme for the approximation of the vectorial modified Korteweg-de Vries equation. We demonstrate that the scheme conserves energy up to machine precision. In this sense the method is consistent with the energy balance of the continuous system. This energy balance ensures there is no numerical dissipation allowing for extremely accurate long time simulations free from numerical artifacts. Various numerical experiments are shown demonstrating the asymptotic convergence of the method with respect to the discretisation parameters. Some simulations are also presented that correctly capture the unusual interactions between solitons in the vectorial setting.

James Jackaman, Oliver Sutton

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

Elena Celledoni, James Jackaman, Davide Murari et al.

As supported by abundant experimental evidence, neural networks are state-of-the-art for many approximation tasks in high-dimensional spaces. Still, there is a lack of a rigorous theoretical understanding of what they can approximate, at which cost, and at which accuracy. One network architecture of practical use, especially for approximation tasks involving images, is (residual) convolutional networks. However, due to the locality of the linear operators involved in these networks, their analysis is more complicated than that of fully connected neural networks. This paper deals with approximation of time sequences where each observation is a matrix. We show that with relatively small networks, we can represent exactly a class of numerical discretizations of PDEs based on the method of lines. We constructively derive these results by exploiting the connections between discrete convolution and finite difference operators. Our network architecture is inspired by those typically adopted in the approximation of time sequences. We support our theoretical results with numerical experiments simulating the linear advection, heat, and Fisher equations.