Cross-term extreme learning machine for accurate reliability analysis

Purpose To address the challenge of balancing accuracy and efficiency in existing neural network-assisted structural reliability analysis methods for low-sample-size and high-dimensional problems, this study proposes a simple and highly efficient deep network model that significantly improves prediction accuracy while retaining the rapid training advantage of extreme learning machine (ELM). Furthermore, the optimal configuration of the model's key parameters is determined, providing a more efficient and accurate technical tool for quantifying uncertainties in engineering structures. Design/methodology/approach This study proposes a structural reliability method based on the cross-term extreme learning machine (CTELM) model. This model adds a cross-term layer to the ELM, calculating cross-terms by pairwise combinations of hidden layer nodes. Least squares estimation (LSE) is used for rapid training, and a considering cross node (CCN) is introduced to balance accuracy and computational efficiency. Furthermore, comparative experiments are conducted using three high-dimensional engineering cases, focusing on the impact of different training sample sizes, the number of hidden nodes and the CCN value on model performance analysis. Findings The results show that CTELM achieves dual optimization in accuracy and efficiency: in scenarios with low training sample size (N < 100), its reliability index error is consistently lower than other models, with an error of only 0.4% when N = 100 in the composite wood-aluminum beam case; its training efficiency is comparable to ELM and superior to artificial neural networks (ANN) and deep neural network, which rely on backpropagation (BP) training, with training time in the cantilever beam case being less than 1/3 of that of ANN. Therefore, the proposed CTELM can achieve high-precision prediction without a large number of training samples, making up for the performance deficiencies of traditional models in low-sample, high-dimensional scenarios. Originality/value The core innovation of this study lies in proposing an improved ELM framework based on a cross-term layer. By adding a cross-term layer between the hidden and output layers, the model's expressive power is expanded through the nonlinear combination of hidden nodes, overcoming the limitation of traditional ELM relying on linear mapping of a single hidden layer. A cross-node selection mechanism (CCN) is designed to avoid the surge in computational cost caused by fully connected cross-terms, balancing model complexity and practicality.