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This article presents a reformulation of the Theory of FunctionalConnections: a general methodology for functional interpolation that can embeda set of user-specified linear constraints. The reformulation presented in thispaper exploits the underlying functional structure presented in the seminalpaper on the Theory of Functional Connections to ease the derivation of theseinterpolating functionals--called constrained expressions--and providesrigorous terminology that lends itself to straightforward derivations ofmathematical proofs regarding the properties of these constrained expressions.Furthermore, the extension of the technique to and proofs in n-dimensions isimmediate through a recursive application of the univariate formulation. Inall, the results of this reformulation are compared to prior work to highlightthe novelty and mathematical convenience of using this approach. Finally, themethodology presented in this paper is applied to two partial differentialequations with different boundary conditions, and, when data is available, theresults are compared to state-of-the-art methods.

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