Bayesian multiscale deep generative model for the solution of high-dimensional inverse problems

•A multiscale generative model is introduced using a variational auto-encoder approach.•A spatial field is computed at various scales with appropriately designed latent space.•Bayesian inference of latent variables is performed from coarse- to fine-scales.•A pre-trained generative model computes parameters of interest at each scale.•Gaussian and channelized permeability estimation in porous media flows is considered.