Augmented variational superposed Gaussian approximation for Fokker–Planck equations with rational polynomial functions

Abstract

Reliable methods for obtaining time-dependent solutions of Fokker–Planck equations are in high demand in the field of non-equilibrium theory. In this paper, we present a new method based on variational superposed Gaussian approximation (VSGA) and Padé approximants. The VSGA obtains time-dependent probability density functions as a superposition of multiple Gaussian distributions. However, a limitation of the VSGA is that the expectation of the drift term with respect to the Gaussian distribution needs to be calculated analytically, which is typically satisfied when the drift term is a polynomial function. When this condition is not met, the VSGA must rely on the numerical integration of the expectation at each step, resulting in huge computational cost. We propose an augmented VSGA (A-VSGA) method that effectively overcomes the limitation of the VSGA by approximating non-linear functions with the Padé approximant. We apply the A-VSGA to two systems driven by chaotic input signals, a stochastic genetic regulatory system and a soft bistable system, whose drift terms are a rational polynomial function and a hyperbolic tangent function, respectively. The numerical calculations show that the proposed method provides accurate results with less computational time than that required for Monte Carlo simulation.