Analytical solution of stochastic differential equation by multilayer perceptron neural network approximation of Fokker-Planck equation

The Fokker-Planck equation is a useful tool to analyze the transient probability density function of the states of a stochastic differential equation. In this paper, a multilayer perceptron neural network is utilized to approximate the solution of the Fokker-Planck equation. To use unconstrained optimization in neural network training, a special form of the trial solution is considered to satisfy the initial and boundary conditions. The weights of the neural network are calculated by Levenberg-Marquardt training algorithm with Bayesian regularization. Three practical examples demonstrate the efficiency of the proposed method.