Bayesian methods for non-linear blind inverse problems

Blind inverse problems (i.e. inverse problems with unknown parameters of the forward model)
are well studied for models with uniform grids, such as blind image deconvolution or blind signal
separation. Recently, new methods of learning of non-linear problems with differentiable
nonlinearities (i.e. neural networks) have been proposed, however they rely on supervised
learning on a training set. The aim of this project is to develop methods of blind inverse
problems with a general observation operator (such as irregular grids) and general parametric
structure that can be learned from a single sample. The main challenge is to find suitable
regularization structure. We propose to use the Bayesian approach defining regularization using
prior distribution and novel methods of distribution representation developed in machine
learning such as variational autoencoders. The proposed methods will be validated in
specialized models for: i) determination of a source term of an atmospheric release, and ii)
adaptive measurement grid refinement in environmental electron microscopy.