The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations

TL;DR

This paper reformulates the Theory of Functional Connections to simplify the derivation of constrained functionals, extends it to multiple dimensions, and applies it to solve partial differential equations with boundary conditions.

Contribution

It introduces a new, mathematically rigorous reformulation of the theory, enabling easier derivation and extension to higher dimensions, and demonstrates its application to PDEs.

Findings

Reformulation simplifies derivation of constrained expressions.

Extension to n-dimensions is straightforward via recursion.

Method achieves competitive results on PDE boundary value problems.

Abstract

This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals--called constrained expressions--and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expressions. Furthermore, the extension of the technique to and proofs in n-dimensions is immediate through a recursive application of the univariate formulation. In all, the results of this reformulation are compared to prior work to highlight the novelty and mathematical convenience of using this approach. Finally, the methodology presented in this paper is applied to two partial differential equations with different boundary conditions, and, when data is available, the results are compared to state-of-the-art methods.