Methodology for the solutions of some reduced Fokker‐Planck equations in high dimensions

TLDR

The study introduces an analytical framework for solving high‑dimensional reduced Fokker‑Planck equations by partitioning the state space of large‑scale nonlinear stochastic systems into two subspaces. By integrating the Fokker‑Planck equation over one subspace and formulating the joint probability density for the remaining subspace, the high‑dimensional problem is reduced to low‑dimensional equations that are solvable using an exponential‑polynomial closure method. Numerical experiments demonstrate that the proposed method outperforms Monte‑Carlo simulation and equivalent linearization, confirming its effectiveness.

Abstract

Abstract In this paper, a new methodology is formulated for solving the reduced Fokker‐Planck (FP) equations in high dimensions based on the idea that the state space of large‐scale nonlinear stochastic dynamic system is split into two subspaces. The FP equation relevant to the nonlinear stochastic dynamic system is then integrated over one of the subspaces. The FP equation for the joint probability density function of the state variables in another subspace is formulated with some techniques. Therefore, the FP equation in high‐dimensional state space is reduced to some FP equations in low‐dimensional state spaces, which are solvable with exponential polynomial closure method. Numerical results are presented and compared with the results from Monte Carlo simulation and those from equivalent linearization to show the effectiveness of the presented solution procedure. It attempts to provide an analytical tool for the probabilistic solutions of the nonlinear stochastic dynamics systems arising from statistical mechanics and other areas of science and engineering.