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

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.

arXiv comment33 pages, 6 figures, 14 tables

References in corpus (6)

• Artificial Neural Networks for Solving Ordinary and Partial Differential Equations
• Deep Theory of Functional Connections: A New Method for Estimating the Solutions of PDEs
• Fuel-Efficient Powered Descent Guidance on Large Planetary Bodies via Theory of Functional Connections
• Analytically Embedding Differential Equation Constraints into Least Squares Support Vector Machines using the Theory of Functional Connections
• Least-squares solutions of boundary-value problems in hybrid systems
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• Theory of Functional Connections and Nelder-Mead optimization methods applied in satellite characterization
• Connecting Earth and Moon via the L1 Lagrangian point