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Differential equations (DEs) are used as numerical models to describephysical phenomena throughout the field of engineering and science, includingheat and fluid flow, structural bending, and systems dynamics. While there aremany other techniques for finding approximate solutions to these equations,this paper looks to compare the application of the Theory of FunctionalConnections (TFC) with one based on least-squares support vector machines(LS-SVM). The TFC method uses a constrained expression, an expression thatalways satisfies the DE constraints, which transforms the process of solving aDE into solving an unconstrained optimization problem that is ultimately solvedvia least-squares (LS). In addition to individual analysis, the two methods aremerged into a new methodology, called constrained SVMs (CSVM), by incorporatingthe LS-SVM method into the TFC framework to solve unconstrained problems.Numerical tests are conducted on four sample problems: One first order linearordinary differential equation (ODE), one first order nonlinear ODE, one secondorder linear ODE, and one two-dimensional linear partial differential equation(PDE). Using the LS-SVM method as a benchmark, a speed comparison is made forall the problems by timing the training period, and an accuracy comparison ismade using the maximum error and mean squared error on the training and testsets. In general, TFC is shown to be slightly faster (by an order of magnitudeor less) and more accurate (by multiple orders of magnitude) than the LS-SVMand CSVM approaches.

Analytically Embedding Differential Equation Constraints into Least Squares Support Vector Machines using the Theory of Functional Connections