Analytically Embedding Differential Equation Constraints into Least Squares Support Vector Machines using the Theory of Functional Connections
This work addresses the problem of efficiently and accurately solving differential equations for engineers and scientists, but it is incremental as it builds on existing methods like TFC and LS-SVM.
The paper tackled solving differential equations by comparing the Theory of Functional Constraints (TFC) with least-squares support vector machines (LS-SVM) and a merged method, showing that TFC was slightly faster (by an order of magnitude or less) and more accurate (by multiple orders of magnitude) in numerical tests on sample problems.
Differential equations (DEs) are used as numerical models to describe physical phenomena throughout the field of engineering and science, including heat and fluid flow, structural bending, and systems dynamics. While there are many other techniques for finding approximate solutions to these equations, this paper looks to compare the application of the Theory of Functional Connections (TFC) with one based on least-squares support vector machines (LS-SVM). The TFC method uses a constrained expression, an expression that always satisfies the DE constraints, which transforms the process of solving a DE into solving an unconstrained optimization problem that is ultimately solved via least-squares (LS). In addition to individual analysis, the two methods are merged into a new methodology, called constrained SVMs (CSVM), by incorporating the LS-SVM method into the TFC framework to solve unconstrained problems. Numerical tests are conducted on four sample problems: One first order linear ordinary differential equation (ODE), one first order nonlinear ODE, one second order linear ODE, and one two-dimensional linear partial differential equation (PDE). Using the LS-SVM method as a benchmark, a speed comparison is made for all the problems by timing the training period, and an accuracy comparison is made using the maximum error and mean squared error on the training and test sets. In general, TFC is shown to be slightly faster (by an order of magnitude or less) and more accurate (by multiple orders of magnitude) than the LS-SVM and CSVM approaches.