CAREER: Learning Kernels in Operators from Data: Learning Theory, Scalable Algorithms and Applications

Learning kernels/functions in operators from data is a new frontier that bridges computational mathematics, inverse problems, statistical inference, and machine learning. Such operator learning problems arise in various applications in disciplines such as physics, biology, engineering, economics, and hydrography. These kernels/functions represent the intrinsic physical laws of interactions between particles or the inherent structures of the operators. Thus, they are as fundamental as the physical laws of gravity in physics, shredding insight into models in their applications. Therefore, it is paramount to construct robust convergent estimators when the data size increases to reveal the intrinsic laws that are not sensitive to the data. This project will develop a unified computational approach for the nonparametric learning of kernels/functions in operators from high- or infinite dimensional data, and the products are scalable algorithms with performance guarantees in a unifying variational framework. It will study four groups of applications where the goal is to recover the kernels/functions in PDE operators, non-Markovian processes, state-space models, and weighted convolution operators. Notably, a systematic learning theory, covering identifiability and regularization, will address the challenges from nonlocal dependence and high-dimensional data. Furthermore, this theory builds the mathematical foundations for a general framework, making the algorithms and theory applicable to inverse problems far beyond those studied in this proposal. Finally, the research will be integrated into undergraduate and graduate teaching in multiple disciplines, promote the fundamental role of computational math in statistical/machine learning, and provide many research opportunities for graduate students. 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.