Optimal Control, Numerics, and Applications of Fractional PDEs

TL;DR

This paper reviews recent advances in fractional PDEs, focusing on operators like fractional Laplacian and fractional time derivatives, their applications, solution notions, numerical schemes, and optimal control problems, including deep learning connections.

Contribution

It provides a comprehensive overview of solution concepts, numerical methods, and optimal control applications for fractional PDEs, highlighting recent developments and open challenges.

Findings

Numerical schemes for fractional Laplacian and time derivatives are discussed.

Optimal control problems with fractional PDE constraints are analyzed.

Connections between fractional PDEs and deep neural networks are explored.

Abstract

This article provides a brief review of recent developments on two nonlocal operators: fractional Laplacian and fractional time derivative. We start by accounting for several applications of these operators in imaging science, geophysics, harmonic maps and deep (machine) learning. Various notions of solutions to linear fractional elliptic equations are provided and numerical schemes for fractional Laplacian and fractional time derivative are discussed. Special emphasis is given to exterior optimal control problems with a linear elliptic equation as constraints. In addition, optimal control problems with interior control and state constraints are considered. We also provide a discussion on fractional deep neural networks, which is shown to be a minimization problem with fractional in time ordinary differential equation as constraint. The paper concludes with a discussion on several open problems.