By measuring responses of physical systems, it is possible to gain knowledge about their properties by inversion or their states by data assimilation. Many natural systems display strong spatial heterogeneity and temporal dynamics acting over many scales. The information content in the available noise-contaminated data is generally insufficient to derive unique estimates at relevant spatial and temporal scales. Deep generative modeling excels at generating (time-dependent) unconditional realizations of properties or states featuring similar characteristics as the training data. This provides a new way to address Bayesian inverse problems (and data assimilation) in which a deep generative model representing prior knowledge is combined with a likelihood term accounting for the available site-specific data to estimate the posterior distribution. Using examples from hydrogeology and geophysics, I will review and discuss some of the more common deep generative models (variational autoencoders, generative adversarial networks, diffusion models) to represent prior knowledge, as well as the strengths and weaknesses of alternative approaches to estimate the posterior distribution (Markov chain Monte Carlo, sequential Monte Carlo, variational Bayes, diffusion posterior sampling). I will end by highlighting the main challenges of such inversion workflows.
