# Augmented Variational Superposed Gaussian Approximation for Langevin Equations with Rational Polynomial Functions

## TL;DR

This paper introduces an augmented variational superposed Gaussian approximation method that efficiently solves Langevin equations with rational polynomial drift functions, reducing computational costs compared to Monte Carlo simulations.

## Contribution

The paper proposes an augmented VSGA method using Padé approximant to handle non-polynomial drift functions in Langevin equations, improving efficiency and accuracy.

## Key findings

- Accurately solves Langevin equations with rational polynomial drift functions.
- Reduces computational cost compared to Monte Carlo methods.
- Successfully applied to chaotic, genetic, and bistable systems.

## Abstract

Reliable methods for obtaining time-dependent solutions of Langevin 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 Pad\'e approximant. 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 should 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 Pad\'e 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 can provide accurate results with less temporal cost than that required for Monte Carlo simulation.