  The combination of machine learning models with physical models is a recent
research path to learn robust data representations. In this paper, we introduce
p$^3$VAE, a generative model that integrates a physical model which
deterministically models some of the true underlying factors of variation in
the data. To fully leverage our hybrid design, we enhance an existing
semi-supervised optimization technique and introduce a new inference scheme
that comes along meaningful uncertainty estimates. We apply p$^3$VAE to the
pixel-wise classification of airborne hyperspectral images. Our experiments on
simulated and real data demonstrate the benefits of our hybrid model against
conventional machine learning models in terms of extrapolation capabilities and
interpretability. In particular, we show that p$^3$VAE naturally has high
disentanglement capabilities. Our code and data have been made publicly
available at https://github.com/Romain3Ch216/p3VAE.
