  Statistical postprocessing is used to translate ensembles of raw numerical
weather forecasts into reliable probabilistic forecast distributions. In this
study, we examine the use of permutation-invariant neural networks for this
task. In contrast to previous approaches, which often operate on ensemble
summary statistics and dismiss details of the ensemble distribution, we propose
networks that treat forecast ensembles as a set of unordered member forecasts
and learn link functions that are by design invariant to permutations of the
member ordering. We evaluate the quality of the obtained forecast distributions
in terms of calibration and sharpness and compare the models against classical
and neural network-based benchmark methods. In case studies addressing the
postprocessing of surface temperature and wind gust forecasts, we demonstrate
state-of-the-art prediction quality. To deepen the understanding of the learned
inference process, we further propose a permutation-based importance analysis
for ensemble-valued predictors, which highlights specific aspects of the
ensemble forecast that are considered important by the trained postprocessing
models. Our results suggest that most of the relevant information is contained
in a few ensemble-internal degrees of freedom, which may impact the design of
future ensemble forecasting and postprocessing systems.
