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Analytical solution of stochastic differential equation by multilayer perceptron neural network approximation of Fokker–Planck equation
Author(s) -
Namadchian Ali,
Ramezani Mehdi
Publication year - 2020
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22445
Subject(s) - fokker–planck equation , artificial neural network , stochastic differential equation , probability density function , differential equation , mathematics , regularization (linguistics) , multilayer perceptron , mathematical optimization , computer science , artificial intelligence , mathematical analysis , statistics
Abstract 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.