a physics-informed lssvr method with legendre kernels for direct solution of fokker-planck equations

The fokkerplanck equation is widely used to describe how systems evolve when randomness plays a role. it appears in many fields, including physics, finance, biology, and engineering. classical numerical methods usually require discretization, which can make the computation expensive, unstable, or less accurate. in this work, we present a direct method for solving these equations using least squares support vector regression (lssvr) with legendre kernels. our approach avoids discretization and provides global optimization, which helps overcome the difficulties faced by loss-based methods such as physics-informed neural networks (pinns). the use of legendre kernels gives strong approximation properties and ensures high accuracy in the solutions. we tested the method on several problems and found that it achieves very precise results while being faster and more stable than pinns. to further improve reliability, we also applied automatic hyperparameter tuning, which adapts the method to each problem without manual adjustment. these results show that lssvr with legendre kernels is a simple, accurate, and efficient tool for scientists and engineers who need to solve fokkerplanck equations.