A data-driven method for the steady state of randomly perturbed dynamics

TL;DR

This paper introduces a data-driven approach to accurately compute the invariant probability density function of randomly perturbed dynamical systems by replacing boundary conditions with a least squares problem based on Monte Carlo simulations.

Contribution

It presents a novel method that replaces traditional boundary conditions with a least squares formulation, enabling high-accuracy solutions in local regions regardless of attractor coverage.

Findings

Achieves high-accuracy density estimation in local areas

Works effectively regardless of attractor coverage

Utilizes Monte Carlo simulations for reference solutions

Abstract

We demonstrate a data-driven method to solve for the invariant probability density function of a randomly perturbed dynamical system. The key idea is to replace the boundary condition of numerical schemes by a least squares problem corresponding to a reference solution, which is generated by Monte Carlo simulation. With this method we can solve for the invariant probability density function in any local area with high accuracy, regardless of whether the attractor is covered by the numerical domain.