Variational autoencoder for the identification of piecewise models

The paper presents a variational autoencoder (VAE) tailored for the identification of hybrid piecewise models in input-output form. We show that using a specialized autoencoder structure, the latent space can provide an interpretable representation in terms of the modes of the underlying hybrid system. In particular, we use categorical encoding of the discrete latent variables whose distribution is approximated via the encoder neural network, characterizing a partition of the regressor space, while the decoder consists of a set of neural networks, each corresponding to a local submodel of the piecewise hybrid system. By employing variational Bayesian framework for inference, the constitutive terms of the evidence lower bound (ELBO) are derived analytically with the chosen VAE architecture. The ELBO loss consists of a reconstruction error term and a regularization term over the latent modes. This loss is optimized in order to train the encoder-decoder networks concurrently via back-propagation. The developed framework is not restricted to simple piecewise affine (PWA) models and it can be straightforwardly extended to general class of piecewise non-linear systems over non-polyhedral domains.