# Optimal Control, Numerics, and Applications of Fractional PDEs

arXiv: https://arxiv.org/abs/2106.13289 · HTML: https://mlsanity.com/paper/2106.13289/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Jun 2021.

## Built on by (most cited first, 1 of 1)

- [OCTANE -- Optimal Control for Tensor-based Autoencoder Network Emergence: Explicit Case](https://mlsanity.com/paper/2509.08169/) (2025)

## Abstract (arXiv)

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.