Solving Fokker Planck Equation Research Articles

Structure‐preserving numerical methods for Fokker–Planck equations

AbstractA common way to numerically solve Fokker–Planck equations is the Chang–Cooper method in space combined with one of the Euler methods in time. However, the explicit Euler method is only conditionally positive, leading to severe restrictions on the time step to ensure positivity. On the other hand, the implicit Euler method is robust but nonlinearly implicit. Instead, we propose to combine the Chang–Cooper method with unconditionally positive Patankar‐type time integration methods, since they are unconditionally positive, robust for stiff problems, only linearly implicit, and also higher‐order accurate. We describe the combined approach, analyze it, and present a relevant numerical example demonstrating advantages compared to schemes proposed in the literature.

A machine learning framework for efficiently solving Fokker–Planck equations

A machine learning framework for efficiently solving Fokker–Planck equations

Deep learning framework for solving Fokker–Planck equations with low-rank separation representation

An insightful deep learning framework is proposed to solve the well-known Fokker–Planck (FP) equations that quantify the evolution of the probability density function. It efficiently reduces the demand of training data in acquiring precise integrations of special normalization conditions via neural network (NN). Instead of all hypercubic discrete points, the inputs of each NN only require one-dimensional discrete data, and this also avoids the exponential increase in training data as the dimension increase. Without loss of generality, to solve a d-dimensional FP equation, d NNs are employed and assembled into a low-rank separation representation. The FP equation, boundary conditions, and integral operators are then re-expressed in the sense of the separation representation. It enables the constructed loss function to perform simple vector operations, in that complicated d-dimensional operators are replaced by a set of one dimensional operators. A tractable strategy is presented for the selection of separation rank inspired by the potential function of the given system, although selecting an appropriate separation rank is still an open issue. Typical numerical examples reveal that the proposed algorithm is effective and superior for solving FP equations. The suggested framework could be applied and extended in various areas of engineering and applied sciences.

Solving Fokker–Planck equations using deep KD-tree with a small amount of data

The Fokker–Planck (FP) equation can deterministically describe the evolution of the probability density function, which plays an extremely significant role in the fields of stochastic dynamics. Unfortunately, the limited samples that arise from the consideration of engineering practice are inevitable, which restricts the solving of the FP equation. Accordingly, in the present study, a super-DL-FP framework is established to solve the steady-state FP equation with a small amount of data, through combining the deep KD-tree and the DL-FP approach proposed in [Chaos 30, 013133 (2020)]. It should be emphasized that the normalization condition is of great importance and has to be considered in solving the steady-state FP equation. An appropriate integral estimation for the normalization condition under non-uniform meshing can effectively improve the precision of the solution, but it is still a challenging problem, especially for the case of small data. Thus, the so-called deep KD-tree method is innovatively proposed to estimate the normalized integral with a small random dataset. The main target is to obtain the appropriate discrete integral points and corresponding integral volumes by executing multiple KD-tree segmentation based on random data on the integral region. Several numerical experiments and comparisons are implemented to illustrate the superior performance of the super-DL-FP method. The obtained results indicate that the proposed algorithm can accomplish higher accuracy in the sense of lower cost than the well-known algorithms like center difference scheme, Chebyshev spectrum algorithm, and normalized flow approach.

Data driven adaptive Gaussian mixture model for solving Fokker-Planck equation.

The Fokker-Planck (FP) equation provides a powerful tool for describing the state transition probability density function of complex dynamical systems governed by stochastic differential equations (SDEs). Unfortunately, the analytical solution of the FP equation can be found in very few special cases. Therefore, it has become an interest to find a numerical approximation method of the FP equation suitable for a wider range of nonlinear systems. In this paper, a machine learning method based on an adaptive Gaussian mixture model (AGMM) is proposed to deal with the general FP equations. Compared with previous numerical discretization methods, the proposed method seamlessly integrates data and mathematical models. The prior knowledge generated by the assumed mathematical model can improve the performance of the learning algorithm. Also, it yields more interpretability for machine learning methods. Numerical examples for one-dimensional and two-dimensional SDEs with one and/or two noises are given. The simulation results show the effectiveness and robustness of the AGMM technique for solving the FP equation. In addition, the computational complexity and the optimization algorithm of the model are also discussed.

Thermal dynamic phase transition of Reissner-Nordstr\xf6m Anti-de Sitter black holes on free energy landscape

We explore the thermodynamics and the underlying kinetics of the van der Waals type phase transition of Reissner-Nordström anti-de Sitter (RNAdS) black holes based on the free energy landscape. We show that the thermodynamic stabilities of the three branches of the RNAdS black holes are determined by the underlying free energy landscape topography. We suggest that the large (small) RNAdS black hole can have the probability to switch to the small (large) black hole due to the thermal fluctuation. Such a state switching process under the thermal fluctuation is taken as a stochastic process and the associated kinetics can be described by the probabilistic Fokker-Planck equation. We obtained the time dependent solutions for the probabilistic evolution by numerically solving Fokker-Planck equation with the reflecting boundary conditions. We also investigated the first passage process which describes how fast a system undergoes a stochastic process for the first time. The distributions of the first passage time switching from small (large) to large (small) black hole and the corresponding mean first passage time as well as its fluctuations at different temperatures are studied in detail. We conclude that the mean first passage time and its fluctuations are related to the free energy landscape topography through barrier heights and temperatures.

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group.

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

Fifth-order finite-difference scheme for Fokker-Planck equations with drift-admitting jumps.

Recently a useful finite-difference scheme was proposed in [Phys. Rev. E 98, 033302 (2018)2470-004510.1103/PhysRevE.98.033302] to solve Fokker-Planck equations with drift-admitting jumps. However, while the scheme is fifth order for the case with smooth drifts, it is only second order for the case with discontinuous drifts. To rectify this, we propose in this paper an improved scheme that achieves a fifth-order convergence rate for the case with drift-admitting jumps. Numerical experiments are also employed to verify the validity of the scheme.

Variational Iteration Algorithm-I with an Auxiliary Parameter for Solving Fokker-Planck Equation

In this paper, variational iteration algorithm-I with an auxiliary parameter is implemented to investigate Fokker-Planck equations. To show the accuracy and reliability of the technique comparisons are made between the variational iteration algorithm-I with an auxiliary parameter and classic variational iteration algorithm-I. The comparison shows that variational iteration algorithm-I with an auxiliary parameter is more powerful and suitable method for solving Fokker-Planck equations. Furthermore, the proposed algorithm can successfully be applied to a large class of nonlinear and linear problems.

Migration of Gyrotactic Micro-Organisms in Water

Understanding the swimming characteristics of micro-organisms is significant for modelling the migration of motile cells and corresponding ecological risk assessments associated with harmful algae in oceans and estuaries. Presented in this paper is an experimental and numerical investigation of swimming characteristics of a typical gyrotactic micro-organism, Heterosigma akashiwo (H. akashiwo) in water, based on the technology of planar laser-induced fluorescence and the finite volume method. Two-dimensional swimming velocity of algal cells are obtained by analyzing cells’ trajectories in the vertical plane, and three-dimensional swimming velocity is reconstructed based on the assumption that cells’ swimming is isotropic in the horizontal plane. Four important parameters are given to reflect the swimming characteristics of gyrotactic cells in still water, including the mean swimming speed (Vs = 146 μm/s), the relative strength of reorientation by gravitational torque to rotational diffusion (λ = 1.96), the time scale of reorientation (B = 5.6 s), and rotational diffusivity (Dr = 0.046 rad2/s). A database of the ambient vorticity, mean swimming velocity and diffusivity tensor is established, by solving Fokker-Planck equation for the probability density function of cells’ swimming under the combined action of gravity, rotational diffusion, and the ambient vorticity. The mean swimming velocity and translational diffusion tensor of H. akashiwo are found to change with the horizontal and vertical vorticity. It is also shown that gyrotactic cells swim in a given direction for a weak horizontal vorticity, in contrast to cells’ tumbling and being trapped for a strong horizontal vorticity.

Numerical solutions of Fokker-Planck equations with drift-admitting jumps

We develop a finite difference scheme based on a grid staggered by flux points and solution points to solve Fokker-Planck equations with drift-admitting jumps. To satisfy the matching conditions at the jumps, i.e., the continuities of the propagator and the probability current, the jumps are set to be solution points and used to divide the solution domain into subdomains. While the values of the probability current at flux points are obtained within each subdomain, the values of its first derivative at solution points are evaluated by using stencils across the subdomains. Several benchmark problems are solved numerically to show the validity of the proposed scheme.

Bayesian filtering based on solving the Fokker-Planck equation

PurposeThis paper aims to propose Bayesian filtering based on solving the Fokker–Planck equation, to improve the accuracy of filtering in non-Gauss case. Nonlinear filtering plays an important role in many science and engineering fields for estimating the state of dynamic system, but the existing filtering algorithms are mainly used for solving the problem of Gauss system.Design/methodology/approachUnder the Bayesian framework, the time update of this filtering is based on solving Fokker–Planck equation, while the measurement update uses the Bayes formula directly. Therefore, this novel algorithm can be applied to nonlinear, non-Gaussian estimation. To reduce the computational complexity due to standard meshing, an adaptive meshing algorithm proposed which includes the coarse meshing, significant domain determination that is generated using extended Kalman filtering and Chebyshev’s inequality theorem, and value assignment for significant domain. Simulations are conducted on a reentry body tracking problem to demonstrate the effectiveness of this novel algorithm.FindingsIn this way, finer grid points can be placed in the regions with high conditional probability density, while the grid points with low conditional probability density can be neglected. The simulation results indicate that the novel algorithm can reduce the computational burden significantly compared to the standard meshing, while achieving similar accuracy.Practical implicationsA novel Bayesian filtering based on solving the Fokker–Planck equation using adaptive meshing is proposed, and the simulations show that algorithm can reduce the computational burden significantly compared to the standard meshing, while achieving similar accuracy.Originality/valueA novel nonlinear filtering based on solving the Fokker–Planck equation is proposed. The novel algorithm is suitable for non-Gauss system, and can achieve similar accuracy compared to the standard meshing with the significant reduction of computational burden.

Ultrafast nonequilibrium ion and electron dynamics of a neon plasma produced by an ultra-intense x-ray pulse

Ultrafast nonequilibrium ion and electron dynamics of a neon plasma produced in the interaction with an ultra-intense x-ray pulse is investigated theoretically. Electron energy distribution function (EEDF) is obtained by solving Fokker–Planck equation, which is implemented self-consistently in a time-dependent rate equation in the framework of detailed-level-accounting approximation. Evolution dynamics of EEDF are presented at a variety of ion density in interaction with x-ray pulses of different laser intensities. Thermalization of free electrons is demonstrated after the x-ray pulses have turned off. The results are compared with two other simplified models, i.e., one is a relaxation model and the second uses the Maxwellian approach. Large discrepancies in the EEDF are found and the effects of detailed treatment of electron dynamics on population distributions are demonstrated and discussed.

Stabilizing synchrony with heterogeneity

Correlation and synchrony between spike trains of neurons can arise from shared input they receive from other neurons, or from direct connections between the neurons. Physiological heterogeneity can destabilize both coupling-induced and shared input-induced synchronization [1]. In the classical models of synchronization, collective state of a system of coupled oscillators is determined by outcome of rivalry between synchronizing effect of connections and desynchronizing effect of inhomogeneity [2]. Yet there are examples of the systems in which inhomogeneity enhances synchrony [3,4]. In this study we have developed a general framework for the correlation of coupled neuronal oscillators with a given phase sensitivity and we have shown that in some cases, for identical neurons, synchronized state is an unstable attractor and arbitrarily weak noise can destroy synchrony. Inhomogeneity in such systems could stabilize synchrony by providing an asymmetric basin of attraction around the stable phase-locked state. This in turn results in a sharper PDF for the time difference between spikes of the two neurons in presence of noise (see Figure ​Figure1A).1A). The analytic results are obtained by solving Fokker-Planck equations for phase oscillators and it is also shown that in presence of stochastic inputs, the most probable phase difference between spike times of the two neurons does not coincide with the stable point of the deterministic equations. Numerical tests on LIF neurons confirm the positive role of inhomogeneity in stabilizing synchrony and increasing correlation of spike trains. The only difference is that for the model neurons with biologically realistic phase response curve (PRC), the time difference between the spikes of two neurons in the stable state increases with inhomogeneity, in the case of LIF neurons, they lock in almost zero phase lag for sufficiently small values of inhomogeneity (see Figure ​Figure1B1B). Figure 1 (A) PDF for the phase difference of two type1 phase oscillators in the presence of noise for three levels of frequency mismatch. Theoretical predictions (solid lines) are in accordance with numerical results shown as histograms. Dash lines show the phase-locked ...

Infinite product expansion of the Fokker-Planck equation with steady-state solution.

We present an analytical technique for solving Fokker–Planck equations that have a steady-state solution by representing the solution as an infinite product rather than, as usual, an infinite sum. This method has many advantages: automatically ensuring positivity of the resulting approximation, and by design exactly matching both the short- and long-term behaviour. The efficacy of the technique is demonstrated via comparisons with computations of typical examples.

Meshless method for the numerical solution of the Fokker–Planck equation

Abstract In this paper numerical meshless method for solving Fokker–Planck equation is considered. This meshless method is based on multiquadric radial basis function and collocation method to approximate the solution. Here we apply θ -weighted finite difference method. The stability analysis of the method is dealt with, using a linearized stability method. Numerical examples illustrate the validity and applicability of the method.

Thermal transport in a 2D stressed nanostructure with mass gradient

Inspired by some recent molecular dynamics (MD) simulations and experiments on suspended graphene nanoribbons, we study a simplified model where the atoms are disposed in a rectangular lattice coupled by nearest neighbor interactions which are quadratic in the interatomic distance. The system has a mechanical strain, and the border atoms are coupled to Langevin thermal baths. Atom masses vary linearly in the longitudinal direction, modeling an isotope or doping distribution. This asymmetry and tension modify thermal properties. Although the atomic interaction is quadratic, the potential is anharmonic in the coordinates. By direct MD simulations and solving Fokker-Planck equations at low temperatures, we can better understand the role of anharmonicities in thermal rectification. We observe an increasing thermal current with an increasing applied mechanical tension. The temperatures and thermal currents vary along the transverse direction. This effect can be useful to establish which parts of the system are more sensitive to thermal damage. We also study thermal rectification as a function of strain and system size.Received: 20 Novembre 2014, Accepted: 17 April 2015; Edited by: C. A. Condat, G. J. Sibona; DOI: http://dx.doi.org/10.4279/PIP.070008Cite as: R Barreto, M F Carusela, A Mancardo Viotti, A G Monastra, Papers in Physics 7, 070008 (2015)This paper, by R Barreto, M F Carusela, A Mancardo Viotti, A G Monastra, is licensed under the Creative Commons Attribution License 3.0.

Monte Carlo method and High Performance Computing for solving Fokker–Planck equation of minority plasma particles

This paper explains how to obtain the distribution function of minority ions in tokamak plasmas using the Monte Carlo method. Since the emphasis is on energetic ions, the guiding-center transformation is outlined, including also the transformation of the collision operator. Even within the guiding-center formalism, the fast particle simulations can still be very CPU intensive and, therefore, we introduce the reader also to the world of high-performance computing. The paper is concluded with a few examples where the presented method has been applied.

New Approaches for Solving Fokker Planck Equation on Cantor Sets within Local Fractional Operators

We discuss new approaches to handling Fokker Planck equation on Cantor sets within local fractional operators by using the local fractional Laplace decomposition and Laplace variational iteration methods based on the local fractional calculus. The new approaches maintain the efficiency and accuracy of the analytical methods for solving local fractional differential equations. Illustrative examples are given to show the accuracy and reliable results.

A note on stochastic semilinear equations and their associated Fokker–Planck equations

The main purpose of this paper is to prove existence and uniqueness of (probabilistically weak and strong) solutions to stochastic differential equations (SDE) on Hilbert spaces under a new approximation condition on the drift, recently proposed in [6] to solve Fokker–Planck equations (FPE), extended in this paper to a considerably larger class of drifts. As a consequence we prove existence of martingale solutions to the SDE (whose time marginals then solve the corresponding FPE). Applications include stochastic semilinear partial differential equations with white noise and a non-linear drift part which is the sum of a Burgers-type part and a reaction diffusion part. The main novelty is that the latter is no longer assumed to be of at most linear, but of at most polynomial growth. This case so far had not been covered by the existing literature. We also give a direct and more analytic proof for existence of solutions to the corresponding FPE, extending the technique from [6] to our more general framework, which in turn requires to work on a suitable Gelfand triple rather than just the Hilbert state space.