The theory of functional connections (TFC) is a mathematical framework for functional interpolation. It provides a method for deriving a functional—a function that operates on another function—which can transform constrained optimization problems into equivalent unconstrained ones. This transformation allows TFC to be applied to a wide range of mathematical problems, including the solution of differential equations. In this context, functional interpolation refers to the construction of functionals that always satisfy specified constraints, regardless of how the internal (or free) function is expressed.
From interpolation to functional interpolation To provide a general context for the TFC, consider a generic interpolation problem involving n {\displaystyle n} constraints, such as a differential equation subject to a boundary value problem (BVP). Regardless of the differential equation, these constraints may be consistent or inconsistent. For instance, in a problem over the domain D : [ 0 , 1 ] × [ 0 , 1 ] {\displaystyle {\mathcal {D}}:[0,1]\times [0,1]} , the constraints f 1 ( x , 0 ) = 1 + x {\displaystyle f_{1}(x,0)=1+x} and f 2 ( 0 , y ) = 2 − y {\displaystyle f_{2}(0,y)=2-y} are inconsistent, as they yield different values at the shared point ( 0 , 0 ) {\displaystyle (0,0)} . If the n {\displaystyle n} constraints are consistent, a function interpolating these constraints can be constructed by selecting n {\displaystyle n} linearly independent basis functions such as monomials, { 1 , x , x 2 , ⋯ , x n − 1 } {\displaystyle \{1,x,x^{2},\cdots ,x^{n-1}\}} . The chosen set of basis functions may or may not be consistent with the given constraints. For instance, the constraints y ( − 1 ) = y ( + 1 ) = 0 {\displaystyle y(-1)=y(+1)=0} and d y d x | x = 0 = 1 {\displaystyle {\dfrac {dy}{dx}}{\bigg |}_{x=0}=1} are inconsistent with the basis functions, { 1 , x , x 2 } {\displaystyle \{1,x,x^{2}\}} , as can be easily verified. If the basis functions are consistent with the constraints, the interpolation problem can be solved, yielding an interpolant—a function that satisfies all constraints. Choosing a different set of basis functions would result in a different interpolant. When an interpolation problem is solved and an initial interpolant is determined, all possible interpolants can, in principle, be generated by performing the interpolation process with every distinct set of linearly independent basis functions consistent with the constraints. However, this method is impractical, as the number of possible sets of basis functions is infinite. This challenge was addressed through the development of the TFC, an analytical framework for performing functional interpolation introduced by Daniele Mortari at Texas A&M University. The approach involves constructing a functional f ( x , g ( x ) ) {\displaystyle f{\big (}\mathbf {x} ,g(\mathbf {x} ){\big )}} that satisfies the given constraints for any arbitrary expression of g ( x ) {\displaystyle g(\mathbf {x} )} , referred to as the free function. This functional, known as the constrained functional, provides a complete representation of all possible interpolants. By varying g ( x ) {\displaystyle g(\mathbf {x} )} , it is possible to generate the entire set of interpolants, including those that are discontinuous or partially defined.
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