Interacting Particle Solutions of Fokker-Planck Equations Through Gradient-Log-Density Estimation

Area of Science:

• Computational Physics
• Statistical Mechanics
• Numerical Analysis

Background:

• Fokker-Planck equations model stochastic systems using probability density functions.
• Analytical solutions are limited, necessitating numerical methods for many applications.
• Existing numerical methods can be computationally intensive or lack precision.

Purpose of the Study:

• To develop an efficient and reliable computational approach for simulating Fokker-Planck equations.
• To introduce a novel statistical estimator for particle interactions.
• To enable particle-based simulations of Fokker-Planck equations in various dimensions.

Main Methods:

• Simulating Fokker-Planck equation time evolution via a mean-field limit of interacting particles.
• Utilizing a novel statistical estimator for the gradient of the logarithm of particle density.
• Comparing performance against direct stochastic simulations.

Main Results:

• The developed method yields more accurate and less fluctuating statistics.
• Performance is superior to direct stochastic simulations with comparable particle numbers.
• The framework facilitates effortless and reliable particle-based simulations.

Conclusions:

• The novel computational framework enables efficient and accurate particle-based simulations of Fokker-Planck equations.
• The proposed gradient-log-density estimator has broader applications, including optimal control.
• This work advances the numerical treatment of stochastic systems.