The exponential-polynomial-closure with updated weights method for solving the stationary Fokker–Planck–Kolmogorov equation

The Fokker–Planck–Kolmogorov (FPK) equation is a fundamental equation in statistical mechanics, governing the evolution of the probability density function (PDF) for nonlinear stochastic dynamical (NSD) systems driven by Gaussian white noise. In this paper, a novel semi-analytical method, termed the exponential-polynomial-closure with updated weights (EPC-UW), is proposed to solve the stationary FPK equation and determine the PDF of responses in NSD systems. The EPC-UW method assumes that the stationary PDF takes an exponential polynomial form and solves it using an optimization procedure with an iteratively updated weighting function. Unlike the conventional optimization-oriented exponential-polynomial-closure (OEPC) method, which fixes the weighting function as the PDF obtained from the equivalent linearization (EQL) method, the EPC-UW approach uses the approximate PDF solution from the previous iteration as the weight at each step. This adaptive strategy yields substantially improved accuracy. The performance of the EPC-UW method is validated by analyzing two NSD systems with known exact solutions and two systems without analytical solutions. The results demonstrate that the PDF obtained by the EPC-UW method accurately captures the non-Gaussian characteristics of the response PDF, such as tail behaviors and bimodal distributions. Additionally, the EPC-UW method exhibits superior accuracy over the conventional OEPC method while achieving significantly improved computational efficiency compared to Monte Carlo simulation (MCS).