title: OPTIMAL CONTROL PROBLEMS CONSTRAINED BY FRACTIONAL PDES AND APPLICATION TO DEEP NEURAL NETWORKS
Creators: name: (:Unkn) Unknown
Publisher: name: George Mason University
Year: 2021

description: Motivated by several applications in geophysics, imaging, machine learning, elasticity, finance, anomalous diffusion, etc., this thesis develops algorithms to solve optimal control problems constrained by fractional partial differential equations (FPDEs). It also introduces a novel fractional derivative based Deep Neural Networks (DNNs) to efficiently tackle inverse problems. Fractional operators have recently emerged as an excellent modeling alternative to their classical counterparts. This success can be attributed to the facts that these operators allow long range interactions (nonlocal), they can account for memory information and finally they enforce less smoothness than their classical counterparts. By exploiting all these features, the thesis develops several novel mathematical tools and algorithms which are of wider interest. For example, the notions of weak and very-weak solutions to fractional elliptic and parabolic problems have been introduced. Existence, and higher regularity, of solutions to fractional Dirichlet problems with measure-valued datum has been established. Moreau-Yosida regularization based algorithms have been introduced to solve fractional state constrained optimal control problems. The thesis begins by introducing, a new notion of optimal control. In particular, in the parabolic setting, we establish that it is possible to have an external optimal control. Recall that the classical models only allow control placement either on the boundary or inside the domain. A complete analysis of the Dirichlet and Robin optimal control problems, with constraints on the control, has been provided and the presented numerical results confirms the theoretical findings. In addition to the control constraints, obstacle type constraints on the PDE solution naturally arise in many different applications. To tackle this, the thesis introduces novel state constrained optimal control problems with fractional PDEs as constraints. One of the key challenges here is that the Lagrange multiplier corresponding to the state constraints is a signed Radon measure, this results in a low regularity for the adjoint solution. A complete analysis for this problem has been provided. This is followed by a Moreau-Yosida regularization based algorithm to solve both the elliptic and parabolic optimal control problems. Here convergence analysis (with rates) of the regularized solutions to the original one is established. Next, a finite element method is introduced and discretization error estimates are established in the elliptic setting. Theoretical results are substantiated by numerical experiments. Finally, in recent years, deep learning has emerged as the method of choice for classification problems. However, its role in physics based modeling and inverse problems has been limited. Some of the key challenges include, vanishing and exploding gradients. The thesis introduces a new DNN which allows connectivity between different layers. The main novelty is the modeling of DNN using fractional time derivatives instead of the standard one. The resulting fractional-DNN is then applied to learn the parameter-to-solution map in parameterized PDEs. Subsequently, this approximation is employed to solve Bayesian inverse problems. A speedup of over 100 times is observed in comparison to the existing approaches.