Heading estimation is vital for the autonomous flight of unmanned aerial vehicles. Magnetometers are typically used for this purpose, but they are not robust to electro-magnetic interferences. As a promising alternative, we investigate the insect-inspired solution of skylight polarization sensing. In particular, we develop a robust polarization compass for azimuth estimation. Two datasets are created - one based on a Mie scattering simulation, and one containing real-world pictures captured with a polarization camera under a variety of weather conditions. We employ the ResNet-18 model, which is trained and tested on both datasets separately. The trained model is robust to different weather conditions, and is able to directly analyze maps in the instrumental plane. The median error on the (mostly cloudy) real-world images of 4.30 degrees makes it a promising new method for the navigational toolkit of UAVs. We publish the real-world polarization dataset as open access data, in order to facilitate improvements by the community.