Numerical solution of two dimensional Fokker—Planck equations

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.

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“…For the third example, we consider a Harmonic oscillator with damping and noise as follows:

\begin{array}{c} italicdx = italicvdt \\ italicdv = ((), - italicKx - γ v) italicdt + \sqrt{2 σ} italicdw ((), t) \end{array} .

The corresponding FPE of the Harmonic oscillator is:

\partial_{t} p (x, v, t) = - \partial_{x} (vp (x, v, t)) + \partial_{c} ((Kx + γ v) p (x, v, t)) + \partial_{italicvv} p (x, v, t) .

…”

Section: Numerical Examplesmentioning

confidence: 99%

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

Namadchian

Ramezani

2019

Numerical Methods Partial

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“…Supposing thatm 0 := δ x0 (x 0 ∈ R 2 ), it is shown in [60] that the solution m to (5.2) has a density, which has the following explicit expression…”

Section: 1mentioning

confidence: 99%

On the Discretization of Some Nonlinear Fokker--Planck--Kolmogorov Equations and Applications

Carlini¹,

Silva²

2018

SIAM J. Numer. Anal.

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In this work, we consider the discretization of some nonlinear Fokker-Planck-Kolmogorov equations. The scheme we propose preserves the non-negativity of the solution, conserves the mass and, as the discretization parameters tend to zero, has limit measure-valued trajectories which are shown to solve the equation. The main assumptions to obtain a convergence result are that the coefficients are continuous and satisfy a suitable linear growth property with respect to the space variable. In particular, we obtain a new proof of existence of solutions for such equations.We apply our results to several examples, including Mean Field Games systems and variations of the Hughes model for pedestrian dynamics.

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“…More information about the accuracy of the approximations made by RBFs can be found in [12,41]. Let x 1 , x 2 , .…”

Section: Introductionmentioning

confidence: 99%

A high order method for numerical solution of time-fractional KdV equation by radial basis functions

Sepehrian

Shamohammadi

2018

Arab. J. Math.

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A radial basis function method for solving time-fractional KdV equation is presented. The Caputo derivative is approximated by the high order formulas introduced in Buhman (Proc. Edinb. Math. Soc. 36:319-333, 1993). By choosing the centers of radial basis functions as collocation points, in each time step a nonlinear system of algebraic equations is obtained. A fixed point predictor-corrector method for solving the system is introduced. The efficiency and accuracy of our method are demonstrated through several illustrative examples. By the examples, the experimental convergence order is approximately 4 − α, where α is the order of time derivative.Mathematics Subject Classification 65D05 · 65M06 · 65N06 · 65N35

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Numerical solution of two dimensional Fokker—Planck equations

Cited by 45 publications

References 2 publications

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

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

On the Discretization of Some Nonlinear Fokker--Planck--Kolmogorov Equations and Applications

A high order method for numerical solution of time-fractional KdV equation by radial basis functions

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