The Theory of Functional Connections: A journey from theory to application

TL;DR

The paper presents the development and application of the Theory of Functional Connections, a methodology for embedding constraints into functions to simplify and improve the efficiency of solving constrained optimization and differential equations.

Contribution

It introduces a general formulation of the constrained expressions, providing rigorous mathematical foundations and demonstrating applications to differential equations and real-time optimal control problems.

Findings

Faster and more accurate numerical schemes for constrained problems

Successful application to ordinary differential equations with diverse examples

Feasibility shown for real-time optimal control solutions

Abstract

The Theory of Functional Connections (TFC) is a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The functionals derived from this method, called \emph{constrained expressions}, analytically satisfy the imposed constraints and can be leveraged to transform constrained optimization problems to unconstrained ones. By simplifying the optimization problem, this technique has been shown to produce a numerical scheme that is faster, more accurate, and robust to poor initialization. The content of this dissertation details the complete development of the Theory of Functional Connections. First, the seminal paper on the Theory of Functional Connections is discussed and motivates the discovery of a more general formulation of the constrained expressions. Leveraging this formulation, a rigorous structure of the constrained expression is produced with associated mathematical definitions, claims, and proofs. Furthermore, the second part of this dissertation explains how this technique can be used to solve ordinary differential equations providing a wide variety of examples compared to the state-of-the-art. The final part of this work focuses on unitizing the techniques and algorithms produced in the prior sections to explore the feasibility of using the Theory of Functional Connections to solve real-time optimal control problems, namely optimal landing problems.