Applications of a Bayesian Approach with Deep Learning to Solving Inverse Problems

Inverse problems are among the most challenging and widespread problems in science today. Inverse problems are commonly ill-conditioned, having potentially infinitely many solutions, and prior assumptions must be made in order to make the solution unique. Techniques for solving inverse problems are typically bespoke, requiring different tools and frameworks for each problem.

In this thesis we use Bayes' rule as a generic solution to a range of inverse problems, coupled with the deep learning technique known as normalising flows to provide models for the prior. Bayes' provides great flexibility -- being easily applicable to a range of inverse problems -- as well as results competitive with, or even exceeding, state-of-the-art techniques in image reconstruction, MRI acceleration and radio astronomy.