A machine learning framework for efficiently solving Fokker–Planck equations

Abstract

This paper addresses the challenge of solving Fokker–Planck equations, which are prevalent mathematical models across a myriad of scientific fields. Due to factors like fractional-order derivatives and non-linearities, obtaining exact solutions to this problem can be complex. To overcome these challenges, our framework first discretizes the given equation using the Crank-Nicolson finite difference method, transforming it into a system of ordinary differential equations. Here, the approximation of time dynamics is done using forward difference or an L1 discretization technique for integer or fractional-order derivatives, respectively. Subsequently, these ordinary differential equations are solved using a novel strategy based on a kernel-based machine learning algorithm, named collocation least-squares support vector regression. The effectiveness of the proposed approach is demonstrated through multiple numerical experiments, highlighting its accuracy and efficiency. This performance establishes its potential as a valuable tool for tackling Fokker–Planck equations in diverse applications.