Analytically Embedding Differential Equation Constraints into Least Squares Support Vector Machines Using the Theory...

Area of Science:

• Numerical analysis
• Computational mathematics
• Applied mathematics

Background:

• Differential equations (DEs) are fundamental numerical models in science and engineering.
• Existing methods for solving DEs include least-squares support vector machines (LS-SVM).
• The Theory of Functional Connections (TFC) offers a novel approach using constrained expressions.

Purpose of the Study:

• To compare the efficacy of the Theory of Functional Connections (TFC) against least-squares support vector machines (LS-SVM) for solving differential equations.
• To introduce and evaluate a hybrid method, constrained SVMs (CSVM), combining TFC and LS-SVM.
• To assess both speed and accuracy of TFC, LS-SVM, and CSVM across various differential equation types.

Main Methods:

• The TFC method reformulates DEs as unconstrained optimization problems solved via least-squares.
• LS-SVM is used as a benchmark for comparison.
• A novel CSVM methodology integrates LS-SVM within the TFC framework.
• Numerical tests were performed on four distinct ODE and PDE problems.

Main Results:

• TFC demonstrated superior accuracy, achieving multiple orders of magnitude lower maximum and mean squared errors.
• TFC was generally faster than LS-SVM and CSVM, with training times an order of magnitude less.
• The CSVM approach showed comparable performance to LS-SVM, indicating TFC's core advantage.

Conclusions:

• TFC provides a more accurate and efficient solution strategy for differential equations compared to LS-SVM and CSVM.
• The TFC method's ability to handle constraints within its framework is key to its improved performance.
• This study highlights TFC as a promising alternative for numerical modeling in engineering and scientific applications.