A deep-learning enhanced probability flow tube method for probability density evolution analysis of nonlinear structures

Abstract

Probability density evolution analysis of nonlinear structures provides a basis for uncertainty propagation and dynamic reliability assessment. The recently developed Probability Flow Tube Method (PFTM) enables the reconstruction of the time-dependent Probability Density Function (PDF) of a quantity of interest without numerically solving the generalized density evolution equation. However, applying PFTM to complex nonlinear structures may require a large ensemble of deterministic response time histories, resulting in considerable computational cost. To address this issue, this study proposes a deep-learning enhanced PFTM framework. A hybrid temporal network is trained on a small number of excitation–response pairs and used to predict a large ensemble of response trajectories for subsequent PDF reconstruction. The predicted trajectories are organized into time-continuous probability flow tubes with time-invariant probability weights. The evolving mean and standard deviation of each tube are determined algebraically by matching characteristic-function Taylor coefficients at the origin, and the global PDF is reconstructed as a probability-weighted mixture of the tube PDFs. The framework is evaluated using a real-world engineering example involving stochastic seismic response analysis of an onshore 5-MW steel–concrete hybrid wind turbine tower. The reconstructed PDF evolution and tail probabilities show close agreement with the corresponding Monte Carlo simulation results obtained using $1\times 10^5$ samples. These results indicate that the deep-learning model can efficiently generate the response ensemble required by PFTM while substantially reducing the number of computationally demanding time–history analyses of nonlinear structures.