Deterministic particle dynamics for simulating Fokker-Planck probability flows

Read here the properly rendered readme file for now...

(under construction - for more detailed info please read the relevant article)

Particle-based framework for simulating solutions of Fokker–Planck equations that

• is effortless to set up
• provides smooth transient solutions
• is computationally efficient.

A. From SDEs to ODEs

• Systems with additive noise

  Consider a stochastic system described by the SDE

  

  The temporal evolution of the probability density of the system state is captured by the Fokker-Planck equation (FPE)

  

  The FPE may be re-written in the form of a Liouville equation

  ! [Eq.(3-5) in the main text]

  

  which in turn may be viewed as an evolution equation of the probability distribution of a statistical ensemble of N deterministic dynamical systems of the form [Eq.(4-5) in the main text]

  

  with i=1,...,N.

• Systems with multiplicative noise

  In a similar vain, for state dependent diffusion

  

  the associated deterministic particle dynamics are

  

  which, by setting become

  ! [Eq.(53) {in the main text] 

  

Eq.(1) and Eq.(2) imply that we may obtain transient solutions of the associated FPEs by simulating ensembles of deterministic trajectories/particles with initial conditions drawn from the starting distribution $p_0(x)$.

However, the deterministic particle dynamics in Eq.(1) and Eq.(2) require the knowledge of $\nabla_x \ln p_t(x)$, i.e. the gradient of the logarithm of the quantity of interest. Enter the gradient-log density estimator (score function estimator)!

B. Gradient-log-density (score function) estimator

C. Smooth transient solutions of Fokker-Planck equations

Citations:

1. Maoutsa, Dimitra; Reich, Sebastian; Opper, Manfred. Interacting Particle Solutions of Fokker–Planck Equations Through Gradient–Log–Density Estimation. Entropy 2020, 22, 802.

2. Hyvärinen, Aapo. Estimation of non-normalized statistical models by score matching. Journal of Machine Learning Research 2005, 695-709.