Advancing stochastic modeling for nonlinear problems: Leveraging the transformation law of probability density

• Presentation of the transformation law of probability density for nonlinear problems. • Introduction of an efficient stochastic modeling approach utilizing the Finite Cell Weight Variation method, resulting in detailed probability density outputs. • Projection of multi-dimensional probability space onto the physical state space. In engineering, uncertainties pervade product lifecycles, presenting significant challenges to design reliability and safety, particularly in safety-sensitive industries such as nuclear. Stochastic simulations, leveraging Monte Carlo Sampling, machine learning, and parallel computing, are indispensable for addressing these uncertainties. However, they often overlook the direct influence of prediction models on predicted probability distributions, compromising both efficiency and accuracy. This paper thoroughly investigates the impact of prediction models on predicted probability distributions, presenting a novel mathematical framework to establish the transformation law of probability density. Additionally, we develop the Finite Cell Weight Variation method based on this transformation law. The proposed method seamlessly integrates prediction models into state probability predictions, enhancing reliability assessments while preserving high levels of accuracy and computational efficiency. We illustrate the method's effectiveness with practical examples and validation using Latin Hypercube Sampling (LHC), where several input variables are statistically determined. Our estimation of the probability of the predicted state closely aligns with results obtained using LHC. Furthermore, we explore the implications of our findings and outline future directions in stochastic simulations aimed at strengthening reliability assessments.