基于BP神经网络和Laplace渐进积分法的结构可靠性计算

传统基于代理模型的可靠性研究大多将抽样方法与代理模型相结合，并假定随机变量相互独立，且没有考虑到代理模型的不确定性对失效概率的影响。本文将反向传播(BP)神经网络和Laplace渐进积分法相结合，提出一种结合代理模型和高次阶矩的可靠性计算方法，称之为BP-Lap法。采用Latin超立方抽样技术，结合学习函数选取样本点，基于函数逼近原理，利用BP网络代理极限状态方程及其梯度向量和Hessian矩阵。利用训练好的BP网络通过Laplace渐进积分法求解失效概率，基于十折交叉验证思想，得到失效概率取值区间。通过四个算例，分别在随机变量相关和不相关的条件下，验证了BP-Lap法的有效性。研究表明：BP-Lap法可以衡量代理模型的不确定性对失效概率的影响，得到失效概率的上、下界；BP-Lap法同时适用于显示和隐式的极限状态方程，对相关随机变量的可靠性问题具有较高精度。

A structural reliability analysis method based on BP Neural Network and Laplace progressive integration method

In traditional reliability theory based on surrogate model, sampling methods are often used to obtain failure probability, the correlation of random variables and the influence of the uncertainty of the surrogate model are often not considered. This paper proposes a reliability analysis method combining (Back propagation) BP neural network and Laplace progressive integration method, which is called BP-Lap method. Latin hypercube sampling method and a learning function are adopted to generate sample points. Based on function approximation theory, the limit state function and its first and second partial derivatives are all approximated by the BP network. The trained BP network is used to solve the failure probability by the Laplace progressive integration method, and ten-fold cross-validation method is used get the failure probability interval. Four numerical examples are used to verify the effectiveness of the BP-Lap method under correlated and uncorrelated random variables respectively. The research shows that BP-Lap method can measure the influence of the uncertainty of the surrogate model on the failure probability, and obtain the the upper and lower bounds of failure probability. BP-Lap method is suitable for both explicit and implicit limit state functions, and has higher accuracy for reliability problems with correlated random variables.