The Theory of Functional Connections on Vector Spaces

TL;DR

This paper generalizes the Theory of Functional Connections (TFC) from fields to vector spaces and arbitrary inputs, broadening its applicability beyond real numbers and fields, with illustrative Python examples.

Contribution

It extends TFC's theoretical framework from fields to vector spaces and general inputs, enabling wider application and understanding.

Findings

TFC applies to vector spaces, not just fields.

Theorems and proofs hold for more general structures.

Open-access Python code demonstrates practical use.

Abstract

The Theory of Functional Connections (TFC) is most often used for constraints over the field of real numbers. However, previous works have shown that it actually extends to arbitrary fields. The evidence for these claims is restricting oneself to the field of real numbers is unnecessary because all of the theorems, proofs, etc. for TFC apply as written to fields in general. Here, that notion is taken a step further, as fields themselves are unnecessarily restrictive. The theorems, proofs, etc. of TFC apply to vector spaces, not just fields. Moreover, the inputs of the functions/constraints do not even need to be restricted to vector spaces; they can quite literally be anything. In this note, the author shows the more general inputs/outputs of the various symbols that comprise TFC. Examples with accompanying, open-access Python code are included to aid the reader's understanding.