p$^3$VAE: a physics-integrated generative model. Application to the pixel-wise classification of airborne hyperspectral images

Hybrid modeling, that is the combination of data-driven and theory-driven modeling, has recently raised a lot of attention. The integration of physical models in machine learning has indeed demonstrated promising properties such as improved interpolation and extrapolation capabilities and increased interpretability [1, 2]. Conventional machine learning models learn correlations, from a training data set, in order to map observations to targets or latent representations, with the hope to generalize to new data. While many different models could perfectly fit the training data, the assumptions made during the learning process (from the model architecture to the learning algorithm itself), sometimes called inductive biases [3, 4], are crucial to obtain high generalization performances. In contrast, hybrid models are partially grounded on deductive biases, i.e. assumptions derived, in our context, from physics models that generalize, by nature, to out-of-distribution data. Therefore, in various fields for which the data distribution is governed by physical laws, such as fluid dynamics, thermodynamics or solid mechanics, hybrid modeling has recently become a hot topic [5, 6, 7].