Modern Optimal Recovery: Enriched, Nonconvex, Dynamical

About This Grant

Numerous industrial and societal sectors depend heavily on the algorithmic automation that has emerged from Artificial Intelligence and Machine Learning. However, many algorithms were originally conceived to work approximately well most of the time, which does not fit the standards for areas critically important to the United States, such as defense, medicine, and transportation, where slight failure, however infrequent, is not an option. The purpose of the project is to further develop a mathematical framework called Optimal Recovery, which provides guarantees for function learning and recovery under realistic modeling assumptions. The results are expected to have implications in any field of science where average-case guarantees are not sufficient and worst-case guarantees are sought. The project also involves training early career mathematicians in computational and data-related topics that lay at the foundation of timely developments in Artificial Intelligence. The planned investigations are intended to widen the scope and applicability of Optimal Recovery on three selected topics. In the first topic, the theory and practice usually centered on functions with single real-valued outputs will be extended to functions with more complicated outputs. In the second topic, common assumptions of convexity on the quantities of interest and on the model sets will be generalized, allowing the results to apply to non-convex models including those utilizing neural networks. In the third topic, the recovery of dynamical objects will be studied, in particular graphs, that extend to timely and more complex models that can deal with data such as destructive observations. State-of-the-art Optimization Theory will naturally play a prominent role throughout. Some parts of the project are inspired by challenges raised at Los Alamos National Laboratory, where novel numerical methods will be tested in these application-oriented settings, which will in turn provide fresh mathematical insight. The computational outcomes will be made publicly available to the scientific community. 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.