Theory of Functional Connections

Introduction

The Theory of Functional Connections (TFC) is a mathematical framework that enables the exact analytical satisfaction of constraints within function approximation problems. TFC is a mathematical framework to perform functional interpolation. Functional interpolation is the generalization of interpolation. For example, interpolation produces a function that passes through two points, while functional interpolation produces a functional representing all functions passing through those two points, including discontinuous and partially defined functions. In general, TFC analytically derives constrained functionals that always satisfy a set of linear constraints. This way, constrained optimization problems subject to linear constraints, such as differential equations, are transformed into unconstrained problems. Developed to address limitations in traditional methods, TFC is primarily applied in fields requiring complex optimization and solution approximation, including physics, engineering, and control theory.

History and Development

TFC, initially called Theory of Connections, was introduced by Daniele Mortari in the seminal work [1] and immediately used [2] to solve linear ordinary differential equations. The theory's development was then expanded in two doctoral dissertations and summarized in a book [3]. By embedding constraints directly into the functional form, TFC achieves a precise, non-iterative approach, making it suitable for real-time applications. The theory's progression included the formalization of its mathematical foundations and applications in areas like control theory, structural analysis, and space trajectory optimization, where accuracy and efficiency are critical.

Theoretical Framework

This section can introduce the fundamental concepts:

1) Explain how TFC transforms constraints into functional expressions that are embedded within solution spaces.

2) Introduce the fundamental equations or theorems that underlie TFC, perhaps summarizing key equations if they’re concise.

3) Explain the method of embedding constraints within approximation functions to yield exact constraint satisfaction.

This is performed by analytically deriving some functionals, called constrained functionals, representing all functions satisfying a set of linear constraints in n-dimensional space. This way, constrained optimization problems subject to linear constraints, such as differential equations, are transformed into unconstrained problems.

Applications

Detail the various fields where TFC is utilized:

- **Engineering and Physics:** Describe applications in areas like fluid dynamics, structural analysis, or space trajectory optimization.

- **Control Theory:** Mention TFC’s impact on control systems, particularly for constraint-handling in feedback mechanisms.

- **Optimization Problems:** Explain its role in constrained optimization problems and other fields such as machine learning, signal processing, and operations research.

Advantages and Limitations

- **Advantages:** Discuss the benefits of TFC, including the potential for higher accuracy, reduced computational costs, and flexibility in constraint types.

- **Limitations:** Address known limitations of TFC, such as any challenges in applying it to certain types of constraints or high-dimensional systems.

Application in neural Networks

Implementations of TFC in neural networks were first proposed by the Deep-TFC framework, then by the X-TFC using an Extreme learning machine, and by the Physics-informed neural networks (PINN). In particular, TFC allowed PINN to overcome the unbalanced gradients problem that often causes PINNs to struggle to accurately learn the underlying differential equation solution.

Future Research Directions

Discuss current open questions in the field and potential future developments:

- **Extensions and Improvements:** Outline ongoing efforts to expand TFC, such as handling new types of constraints or integrating it with other optimization frameworks.

- **Emerging Applications:** Mention promising new fields where TFC could be applied, from robotics to artificial intelligence.