  In recent years, the field of machine learning has made phenomenal progress
in the pursuit of simulating real-world data generation processes. One notable
example of such success is the variational autoencoder (VAE). In this work,
with a small shift in perspective, we leverage and adapt VAEs for a different
purpose: uncertainty quantification in scientific inverse problems. We
introduce UQ-VAE: a flexible, adaptive, hybrid data/model-informed framework
for training neural networks capable of rapid modelling of the posterior
distribution representing the unknown parameter of interest. Specifically, from
divergence-based variational inference, our framework is derived such that most
of the information usually present in scientific inverse problems is fully
utilized in the training procedure. Additionally, this framework includes an
adjustable hyperparameter that allows selection of the notion of distance
between the posterior model and the target distribution. This introduces more
flexibility in controlling how optimization directs the learning of the
posterior model. Further, this framework possesses an inherent adaptive
optimization property that emerges through the learning of the posterior
uncertainty.
