Operator learning meets inverse problems

Data Science Seminar

Nicholas Nelsen
Cornell University

Abstract

Operator learning involves data-driven models that accept continuum function data as inputs or outputs. Such models are robust to refinement of numerical discretizations and are thus well-suited for solving problems in computational mathematics. This talk showcases recent efforts to bring operator learning ideas to the field of inverse problems. The focus is on end-to-end learning of inverse problem solution operators: directly mapping noisy measurements to unknown parameter fields. However, instability with respect to perturbations of the measurements is a fundamental barrier to successful estimation. Focusing on electrical impedance tomography (EIT) as a case study, a theoretical analysis delivers universal approximation guarantees in the presence of noisy boundary measurements. The talk shows that popular and practical neural operator architectures and EIT setups satisfy the required theoretical assumptions. Numerical evidence supports these results with high quality reconstructions.