Support Vector Regression Based Method to Solve Boundary Value Problems

Boundary value problems are a hot research area in the field of equation problems, with the study of boundary value problems for ordinary differential equations being quite mature. However, there is significant research space in solving boundary value problems when the known conditions are discrete points rather than given functions. Support Vector Regression (SVR) is a machine learning method based on statistical learning theory, which shows unique advantages in approximation problems by minimizing empirical risk while ensuring generalization capability. Therefore, this paper combines regularization, reproducing kernel theory, and SVR to investigate boundary value problems. Treating the boundary value problem as an operator equation problem, the relationship between the solution of the equation and the known conditions is obtained using the properties of reproducing kernel spaces. The problem is then transformed into an approximation problem, regularized into a quadratic programming problem, and solved using SVR to obtain a sparse solution composed of support vectors. Error analysis of the numerical solution obtained is conducted using norms in Sobolev spaces, providing an upper bound for the error between the numerical solution and the analytical solution. Taking a second-order three-point boundary value problem as an example, where only discrete values are given as known conditions for solving the equation, experimental results demonstrate that this method outperforms traditional reproducing kernel methods and W-POAFD methods, confirming its high accuracy and effectiveness.