4,629 research outputs found
Parametric Fokker-Planck equation
We derive the Fokker-Planck equation on the parametric space. It is the
Wasserstein gradient flow of relative entropy on the statistical manifold. We
pull back the PDE to a finite dimensional ODE on parameter space. Some
analytical example and numerical examples are presented
Neural Parametric Fokker-Planck Equations
In this paper, we develop and analyze numerical methods for high dimensional
Fokker-Planck equations by leveraging generative models from deep learning. Our
starting point is a formulation of the Fokker-Planck equation as a system of
ordinary differential equations (ODEs) on finite-dimensional parameter space
with the parameters inherited from generative models such as normalizing flows.
We call such ODEs neural parametric Fokker-Planck equation. The fact that the
Fokker-Planck equation can be viewed as the -Wasserstein gradient flow of
Kullback-Leibler (KL) divergence allows us to derive the ODEs as the
constrained -Wasserstein gradient flow of KL divergence on the set of
probability densities generated by neural networks. For numerical computation,
we design a variational semi-implicit scheme for the time discretization of the
proposed ODE. Such an algorithm is sampling-based, which can readily handle
Fokker-Planck equations in higher dimensional spaces. Moreover, we also
establish bounds for the asymptotic convergence analysis of the neural
parametric Fokker-Planck equation as well as its error analysis for both the
continuous and discrete (forward-Euler time discretization) versions. Several
numerical examples are provided to illustrate the performance of the proposed
algorithms and analysis
Influence of the Fluctuations of the Temperature on the Quasi Equilibrium Condensation
Based on the three parametric Lorenz system a model, that allows to describe in a self-consistent way the behavior of the plasma-condensate system near phase equilibrium, was developed. Considering the influence of the fluctuations of the growing surface temperature the evolution equation and the corresponding Fokker-Planck equation were obtained. The phase diagram, which determined the system parameters corresponding to the regime of the porous structure formation, was built
Statistical deconvolution of the free Fokker-Planck equation at fixed time
We are interested in reconstructing the initial condition of a non-linear
partial differential equation (PDE), namely the Fokker-Planck equation, from
the observation of a Dyson Brownian motion at a given time . The
Fokker-Planck equation describes the evolution of electrostatic repulsive
particle systems, and can be seen as the large particle limit of correctly
renormalized Dyson Brownian motions. The solution of the Fokker-Planck equation
can be written as the free convolution of the initial condition and the
semi-circular distribution. We propose a nonparametric estimator for the
initial condition obtained by performing the free deconvolution via the
subordination functions method. This statistical estimator is original as it
involves the resolution of a fixed point equation, and a classical
deconvolution by a Cauchy distribution. This is due to the fact that, in free
probability, the analogue of the Fourier transform is the R-transform, related
to the Cauchy transform. In past literature, there has been a focus on the
estimation of the initial conditions of linear PDEs such as the heat equation,
but to the best of our knowledge, this is the first time that the problem is
tackled for a non-linear PDE. The convergence of the estimator is proved and
the integrated mean square error is computed, providing rates of convergence
similar to the ones known for non-parametric deconvolution methods. Finally, a
simulation study illustrates the good performances of our estimator
Brownian motion ensembles and parametric correlations of the transmission eigenvalues: Application to coupled quantum billiards and to disordered wires
The parametric correlations of the transmission eigenvalues of a
-channel quantum scatterer are calculated assuming two different Brownian
motion ensembles. The first one is the original ensemble introduced by Dyson
and assumes an isotropic diffusion for the -matrix. The second Brownian
motion ensemble assumes for the transfer matrix an isotropic diffusion
yielded by a multiplicative combination law. We review the qualitative
differences between transmission through two weakly coupled quantum dots and
through a disordered line and we discuss the mathematical analogies between the
Fokker-Planck equations of the two Brownian motion models.Comment: 33 pages, 7 postscript figures, the presented abstract is shortened
in comparison to the abstract of the pape
Investigation of nanoporous material under quasi-equilibrium conditions
Based on the three parametric Lorenz system, a model was developed that
permits to describe the behavior of the plasma-condensate system near phase
equilibrium in a self-consistent way. Considering the effect of fluctuations of
the growth surface temperature, the evolution equation and the corresponding
Fokker-Planck equation were obtained. The phase diagram is built which
determines the system parameters corresponding to the regime of the porous
structure formation.Comment: 8 pages, 3 figure
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