A Rayleigh–Ritz formulation of the Wiener path integral technique for high-dimensional nonlinear systems

A novel Rayleigh–Ritz-based formulation of the Wiener path integral (WPI) technique is developed for efficiently determining the stochastic response of high-dimensional nonlinear systems while circumventing the related “curse of dimensionality.” Specifically, based on the standard WPI formulation, the system response transition joint probability density function (PDF) is expressed as a functional integral over all paths satisfying the prescribed initial and final conditions of the response process. Typically, this functional integral is approximated based on the most probable path, which is obtained by solving a functional minimization problem that takes the form of Euler–Lagrange equations. While recent variational formulations allow direct computation of lower-dimensional PDFs, they require cumbersome symbolic-numerical implementations. To address this challenge, an alternative methodology is developed herein that transforms the original functional minimization problem into a standard finite-dimensional optimization problem, readily solvable with various well-established numerical schemes. In this regard, the need for deriving and solving Euler–Lagrange equations with complex boundary conditions is eliminated. Moreover, high-performance computing strategies, including GPU parallelization, are employed to further enhance the computational efficiency of the technique. A representative example is considered relating to a 1000-DOF nonlinear nano-mechanical system under stochastic excitation. The estimated response PDFs based on the WPI technique show excellent agreement with Monte Carlo simulations. These results highlight that the developed technique can accurately and efficiently handle truly high-dimensional nonlinear systems, paving the way for practical probabilistic analyses and uncertainty quantification in complex engineering applications.