Role of Nonlocal Operators in Inverse Problems and Deep Learning

Ratna Khatri, George Mason University

Abstract:  In this talk, we will discuss the applications of nonlocal operators in three kinds of problems. At first we introduce an external source identification problem with fractional partial differential equation (PDE) as constraints. Our motivation to introduce this new class of inverse problems stems from the fact that the classical PDE models only allow the source/control to be placed on the boundary or inside the observation domain where the PDE is fulfilled. Our new approach allows us to place the source/control outside and away from the observation domain. The second problem is motivated by imaging science where we propose to use the fractional Laplacian as a regularizer. In addition, we create a bilevel optimization neural network (BONNet) to learn the optimal regularization parameters, like the strength of regularization and the fractional exponent. As our model problem, we consider tomographic reconstruction and show an improvement in the reconstruction quality, especially for limited data. Lastly, we introduce a novel deep residual neural network with a rigorous mathematical framework. Our network allows information passage across all the network layers.