Machine Learning for Bayesian Inverse Problems

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Artificial intelligence (AI) has had a profound impact in information technology and commerce to the degree of influencing societal transformation over the last decade. The remarkable success of AI has led to a torrent of research aiming to use AI to solve challenging problems in science and engineering. Despite the empirical success of these methods, our mathematical understanding of the underlying algorithms is limited and we do not fully understand why and how these algorithms perform so well, making their predictability less certain. The aim of this project is to address these shortcomings as they pertain to AI algorithms for a large family of engineering problems called "inverse problems", where an unknown parameter is predicted from indirect measurements, such as MRI imaging. The project will also involve outreach activities organized through the University of Washington such as the training and retention of young researchers including explicit plans to involve underrepresented groups in the STEM fields.Recent advances in Machine Learning (ML) have led to the development of novel techniques for the solution of inverse problems but our theoretical understanding of these methods is limited and the majority of them are incapable of uncertainty quantification. The purpose of this project is to address these shortcomings by developing foundational theory for ML methods for Bayesian inverse problems and to design novel computational techniques that enable ML methods to quantify uncertainties. A measure theoretic interpretation of BIPs will be employed in our theoretical analysis to address the questions of well-posedness, stability, and consistency. New computational techniques will be developed using Markov chain Monte Carlo algorithms, data-driven construction of prior information, and recent variational inference techniques based on generative modeling and optimal transport.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.