  The success of deep learning techniques over the last decades has opened up a
new avenue of research for weather forecasting. Here, we take the novel
approach of using a neural network to predict probability density functions
rather than a single output value, thus producing a probabilistic weather
forecast. This enables the calculation of both uncertainty and skill metrics
for the neural network predictions, and overcomes the common difficulty of
inferring uncertainty from these predictions.
  This approach is purely data-driven and the neural network is trained on the
WeatherBench dataset (processed ERA5 data) to forecast geopotential and
temperature 3 and 5 days ahead. An extensive data exploration leads to the
identification of the most important input variables, which are also found to
agree with physical reasoning, thereby validating our approach. In order to
increase computational efficiency further, each neural network is trained on a
small subset of these variables. The outputs are then combined through a
stacked neural network, the first time such a technique has been applied to
weather data. Our approach is found to be more accurate than some numerical
weather prediction models and as accurate as more complex alternative neural
networks, with the added benefit of providing key probabilistic information
necessary for making informed weather forecasts.
