Feynman–Kac-RBFNN for the stochastic analysis of Bouc–Wen hysteretic systems

Abstract

Hysteretic systems are fundamental in engineering, yet their stochastic response analysis remains a significant challenge due to the additional partial differential equations (PDEs) introduced by hysteresis. This paper investigates the application of a Feynman–Kac-based Radial Basis Function Neural Network (RBFNN) method for analyzing the stochastic response of Bouc–Wen hysteretic systems. By reformulating the Fokker–Planck (FPK) equation via the Feynman–Kac framework, the PDE is transformed into an integral form, eliminating high-order derivative computations typical in conventional RBFNN approaches. Additionally, short-time Gaussian approximation (STGA) is employed to derive analytical expressions for stochastic expectations. Building on existing RBFNN techniques, this study incorporates an inscribed spherical sampling strategy to efficiently solve for the trial solution weights. Numerical experiments on Bouc–Wen systems with both softening and hardening characteristics validate the method, showing strong agreement with Monte Carlo Simulations (MCS) in both marginal and joint probability density functions (PDFs). The results reveal bimodal distributions in the softening case, highlighting complex non-Gaussian stochastic dynamics. The framework reduces computational cost while maintaining considerable accuracy, offering a practical and efficient approach for nonlinear stochastic dynamics in engineering applications.