Numerical weather prediction models provide physically consistent forecasts of the state of the atmosphere. Although observations are taken into account in the process of data assimilation, the surface variables of these models are not calibrated for individual weather stations, especially in mountainous terrain. To obtain such calibrated forecasts, statistical post-processing of the model data is required, which establishes a link between the model output and the station observations. There is still debate on whether non-linear interactions play an essential role in this link. Artificial neural networks, a class of statistical models that have recently achieved record results in many research areas, can capture these non-linearities. Despite the major improvements that lead to these outstanding results, neural networks have not been used extensively for statistical post-processing of weather prediction models in recent years. To assess its potential, I compared a feed-forward neural network with a generalized linear model, the regularized logistic regression, on a classical post-processing task: forecasting 3-hourly probability of precipitation at Innsbruck Airport for lead times (forecast horizons) between 3 and 144 hours, given bi-linearly interpolated model data of the European Centre for Medium-Range Weather Forecasts between 2012 and 2017. The neural network outperforms the logistic regression by (1.0 +- 0.1)% (Brier skill score averaged over all lead times) but takes about 220 times longer to train. If the size of the training dataset is reduced from five years to one year, the neural network is still slightly more accurate than the logistic regression. The predictions of the two models are well correlated with each other with a correlation coefficient of 0.974. The ten predictor variables that are most important for the neural network are either directly related to precipitation or the formation and presence of clouds. Seven of these variables are also among the most important variables for the logistic regression. Due to the rather low performance gain compared with logistic regression and the considerably higher effort for tuning, training, and interpretation of the neural network, I conclude that, for the given set-up, neural networks are less suited than regularized logistic regression models. With more training data, more expertise in tuning and training the model and more sophisticated architectures, neural networks may, however, remain a promising option for statistical post-processing of numerical weather prediction models.