Editing Interpolation (section)

==Functional interpolation==
The [[Theory of functional connections|Theory of Functional Connections]] (TFC) is a mathematical framework specifically developed for [https://www.mdpi.com/journal/mathematics/sections/functional_interpolation functional interpolation]. Given any interpolant that satisfies a set of constraints, TFC derives a functional that represents the entire family of interpolants satisfying those constraints, including those that are discontinuous or partially defined. These functionals identify the subspace of functions where the solution to a constrained optimization problem resides. Consequently, TFC transforms constrained optimization problems into equivalent unconstrained formulations. This transformation has proven highly effective in the solution of [[Differential equation|differential equations]]. TFC achieves this by constructing a constrained functional (a function of a free function), that inherently satisfies given constraints regardless of the expression of the free function. This simplifies solving various types of equations and significantly improves the efficiency and accuracy of methods like [[Physics-informed neural networks|Physics-Informed Neural Networks]] (PINNs). TFC offers advantages over traditional methods like [[Lagrange multiplier|Lagrange multipliers]] and [[spectral method]]s by directly addressing constraints analytically and avoiding iterative procedures, although it cannot currently handle inequality constraints.