  Errors in the representation of clouds in convection-permitting numerical
weather prediction models can be introduced by different sources. These can be
the forcing and boundary conditions, the representation of orography, the
accuracy of the numerical schemes determining the evolution of humidity and
temperature, but large contributions are due to the parametrization of
microphysics and the parametrization of processes in the surface and boundary
layers. These schemes typically contain several tunable parameters that are
either not physical or only crudely known, leading to model errors.
Traditionally, the numerical values of these model parameters are chosen by
manual model tuning. More objectively, they can be estimated from observations
by the augmented state approach during the data assimilation. Alternatively, in
this work, we look at the problem of parameter estimation through an artificial
intelligence lens by training two types of artificial neural networks (ANNs) to
estimate several parameters of the one-dimensional modified shallow-water model
as a function of the observations or analysis of the atmospheric state. Through
perfect model experiments, we show that Bayesian neural networks (BNNs) and
Bayesian approximations of point estimate neural networks (NNs) are able to
estimate model parameters and their relevant statistics. The estimation of
parameters combined with data assimilation for the state decreases the initial
state errors even when assimilating sparse and noisy observations. The
sensitivity to the number of ensemble members, observation coverage, and neural
network size is shown. Additionally, we use the method of layer-wise relevance
propagation to gain insight into how the ANNs are learning and discover that
they naturally select only a few gridpoints that are subject to strong winds
and rain to make their predictions of chosen parameters.
