  Recently, there has been a surge of research on data-driven weather
forecasting systems, especially applications based on convolutional neural
networks (CNNs). These are usually trained on atmospheric data represented on
regular latitude-longitude grids, neglecting the curvature of the Earth. We
assess the benefit of replacing the standard convolution operations with an
adapted convolution operation which takes into account the geometry of the
underlying data (Spherenet convolution), specifically near the poles.
Additionally, we assess the effect of including the information that the two
hemispheres of the Earth have "flipped" properties - for example cyclones
circulating in opposite directions - into the structure of the network. Both
approaches are examples of physics-informed machine learning. The methods are
tested on the WeatherBench dataset, at a resolution of $\sim1.4^{\circ}$ which
is higher than many previous studies on CNNs for weather forecasting. For most
lead times up to day +10 for 500 hPa geopotential and 850 hPa temperature, we
find that using Spherenet convolution or including hemisphere-specific
information individually lead to improvement in forecast skill. Combining the
two methods typically gives the highest forecast skill. Our version of
Spherenet is implemented flexibly and scales well to high resolution datasets,
but is still significantly more expensive than a standard convolution
operation. Finally, we analyze cases with high forecast error. These occur
mainly in winter, and are relatively consistent across different training
realizations of the networks, pointing to flow-dependent atmospheric
predictability.
