# The Theory of Connections. Connecting Points

arXiv: https://arxiv.org/abs/1702.06862 · HTML: https://mlsanity.com/paper/1702.06862/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Feb 2017.

## Built on by (most cited first, 10 of 10)

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- [A Unified Hard-Constraint Framework for Solving Geometrically Complex PDEs](https://mlsanity.com/paper/2210.03526/) (2022)
- [AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regression](https://mlsanity.com/paper/2312.14237/) (2023)
- [Quantification of total uncertainty in the physics-informed reconstruction of CVSim-6 physiology](https://mlsanity.com/paper/2408.07201/) (2024)
- [Gated X-TFC: Soft Domain Decomposition for Forward and Inverse Problems in Sharp-Gradient PDEs](https://mlsanity.com/paper/2510.01039/) (2025)
- [Physics-Informed Functional Link Constrained Framework with Domain Mapping for Solving Bending Analysis of an Exponentially Loaded Perforated Beam](https://mlsanity.com/paper/2604.07025/) (2026)
- [Solving Functional Optimization with Deep Networks and Variational Principles](https://mlsanity.com/paper/2410.06277/) (2024)
- [Exact Constraint Enforcement in Physics-Informed Extreme Learning Machines using Null-Space Projection Framework](https://mlsanity.com/paper/2601.10999/) (2026)
- [Comparative Study of Bending Analysis using Physics-Informed Neural Networks and Numerical Dynamic Deflection in Perforated nanobeam](https://mlsanity.com/paper/2604.24768/) (2026)

## Abstract (arXiv)

This study introduces a procedure to obtain general expressions, y = f(x), subject to linear constraints on the function and its derivatives defined at specified values. These constrained expressions can be used describe functions with embedded specific constraints. The paper first shows how to express the most general explicit function passing through a single point in three distinct ways: linear, additive, and rational. Then, functions with constraints on single, two, or multiple points are introduced as well as those satisfying relative constraints. This capability allows to obtain general expressions to solve linear differential equations with no need to satisfy constraints (the "subject to:" conditions) as the constraints are already embedded in the constrained expression. In particular, for expressions passing through a set of points, a generalization of the Waring's interpolation form, is introduced. The general form of additive constrained expressions is introduced as well as a procedure to derive its coefficient functions, requiring the inversion of a matrix with dimensions as the number of constraints.