# Bijective Mapping Analysis to Extend the Theory of Functional Connections to Non-rectangular 2-dimensional Domains

arXiv: https://arxiv.org/abs/2008.07310 · HTML: https://mlsanity.com/paper/2008.07310/

## Key facts

- Cited by 2 papers in the map (data through 7 Oct 2026).
- First posted on arXiv 28 Jul 2020.

## Built on by (most cited first, 2 of 2)

- [Extremization to Fine Tune Physics Informed Neural Networks for Solving Boundary Value Problems](https://mlsanity.com/paper/2406.05290/) (2024)
- [Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework](https://mlsanity.com/paper/2601.10999/) (2026)

## Abstract (arXiv)

This work presents an initial analysis of using bijective mappings to extend the Theory of Functional Connections to non-rectangular two-dimensional domains. Specifically, this manuscript proposes three different mappings techniques: a) complex mapping, b) projection mapping, and c) polynomial mapping. In that respect, an accurate least-squares approximated inverse mapping is also developed for those mappings having no closed-form inverse. The advantages and disadvantages of using these mappings are highlighted and a few examples are provided. Additionally, the paper shows how to replace boundary constraints expressed in terms of a piecewise sequence of functions with a single function, that is compatible and required by the Theory of Functional Connections already developed by rectangular domains.