Theory of functional connections
This transformation enables the application of TFC to various mathematical challenges, including the solution of differential equations.
To provide a general context for the TFC, consider a generic interpolation problem involving
constraints, such as a differential equation subject to a boundary value problem (BVP).
However, this method is impractical, as the number of possible sets of support 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.
, it is possible to generate the entire set of interpolants, including those that are discontinuous or partially defined.
This functional defines the subspace of functions that inherently satisfy the given constraints, effectively reducing the solution space to the region where solutions to the constrained optimization problem are located.
This reformulation allows for simpler and more efficient solution methods, often improving accuracy, robustness, and reliability.
Within this context, the Theory of Functional Connections (TFC) provides a systematic framework for transforming constrained problems into unconstrained ones, thereby streamlining the solution process.
TFC addresses univariate constraints involving points, derivatives, integrals, and any linear combination of these.
[2] The theory is also extended to accommodate infinite and multivariate constraints and applied to solving ordinary, partial, and integro-differential equations.
[3] This includes the consistency challenges associated with boundary conditions that involve shear and mixed derivatives.
is free to take on any arbitrary values beyond the specified constraints, thanks to the infinite flexibility provided by
This universality illustrates how TFC performs functional interpolation: it constructs a function that satisfies the given constraints while simultaneously allowing complete freedom in behavior elsewhere through the choice of
captures all possible functions that meet the given constraints, showcasing the power and generality of TFC in handling a wide variety of interpolation problems.
TFC has been extended and employed in various applications, including its use in shear-type and mixed derivative problems, the analysis of fractional operators,[4] the determination of geodesics for BVP in curved spaces,[5] and in continuation methods.
[6][7] Additionally, TFC has been applied to indirect optimal control,[8][9] the modeling of stiff chemical kinetics,[10] and the study of epidemiological dynamics.
[11] TFC extends into astrodynamics[1], where Lambert's problem is efficiently solved.
[12] It has also demonstrated potential in nonlinear programming[13] and structural mechanics[14][15] and radiative transfer,[16] among other areas.
Of particular note is the application of TFC in neural networks, where it has shown exceptional efficiency,[17][18][19] especially addressing high-dimensional problems and in enhancing the performance of physics-informed neural networks[20] by effectively eliminating constraints from the optimization process, a challenge that traditional neural networks often struggle to address.
This capability significantly improves computational efficiency and accuracy, enabling the resolution of complex problems with greater ease, as proved by the University of Arizona.
[21][22] TFC has been employed with physics-informed neural networks and symbolic regression techniques[23] for physics discovery of dynamical systems.
[24][25] At first glance, TFC and spectral methods may appear similar in their approach to solving constrained optimization problems.
The Lagrange multipliers method is a widely used approach for imposing constraints in an optimization problem.
This technique introduces additional variables, known as multipliers, which must be computed to enforce the constraints.
While the computation of these multipliers is straightforward in some cases, it can be challenging or even practically infeasible in others, thereby adding significant complexity to the problem.
In contrast, TFC doesn't add new variables and enables the derivation of constrained functionals without encountering insurmountable difficulties.
However, it is important to note that the Lagrange multiplier method has the advantage of handling inequality constraints, a capability that TFC currently lacks.
Consequently, supplementary verification procedures or alternative methods may be required to assess and confirm the quality and global validity of the obtained solution.
In summary, while TFC does not entirely replace the Lagrange multipliers method, it serves as a powerful alternative in cases where the computation of multipliers becomes excessively complex or infeasible, provided the constraints are limited to equalities.
