Abstract: When there are correlated variables in the structure, the problem of solving structural reliability becomes very complex. One way to deal with correlation problems is to convert correlated variables into independent variables. The commonly used methods include the Nataf transform, Rackwitz-Fiessler transform, and Rosenblatt transform. When the distribution type of variables or the joint distribution function between variables does not follow a Gaussian distribution, the Nataf transformation method will have significant computational errors. In the process of Rackwitz Feessler transformation, it is necessary to assume that the correlation between variables remains unchanged, or to ignore changes in correlation. This processing method will result in significant calculation errors in the calculation results.
Under the condition that the joint distribution function of variables is known, Rosenblatt transformation is transformed by using the conditional density function of variables and the marginal probability density function. In the process of Rosenblatt transformation, the conditional probability density function can accurately reflect the correlation between variables. Therefore, this transformation is not affected by the distribution type of variables and whether the correlation is linear. It is an accurate transformation method. In the process of Rosenblatt transformation, it is necessary to know the joint distribution function of random variables. However, in practical engineering, it is difficult to obtain the joint distribution function of random variables. Therefore, it brings great limitations to the application of Rosenblatt transformation.
It is difficult to solve the joint probability density function or conditional cumulative distribution function of variables in Rosenblatt transformation. This article proposes a Rosenblatt transformation method based on Copula function and dual neural network. The copula function is introduced to construct the joint probability density function of correlation variables. The selection of the optimal Copula function can be based on AIC or BIC criteria. In addition, we construct a dual neural network model to solve the integration equation. One of the neural networks learns the integrand part of the integral equation. It is called an integrand network. Based on the specific connection with the integrand network in terms of weight and activation function, another neural network is used to construct the original function of the integrand in the integral formula. It is called a primitive function network.
The primitive function network is a three-layer neural network composed of input layer, hidden layer, and output layer. By taking the derivative of the variables in the functional relationship of the original function network, the vector form of the functional relationship between the output and input variables of the integrand network can be obtained. In addition, the integrand network is also a three-layer neural network. In order to improve the computational efficiency of the model, dsigmoid and sigmoid are used as the activation function of the integrand network and the original function network respectively. In the integrand network, variables are divided into N equal parts within the interval and crossed to form input samples for learning. According to the relationship between network parameters in dual neural networks, the network parameters of the original function network can be calculated. Thus, the original function network of the dual neural network is obtained. This achieves integration of variables. Furthermore, the solution of the conditional cumulative distribution function can be achieved. This method breaks the limitations of Rosenblatt transformation in solving structural reliability. It expands the scope of use of the Rosenblatt transformation method.
We apply this method to the reliability calculation of steel beam structures, foundation pile structures, and infinite slope structures. From the calculation results of the examples, it can be seen that the method proposed in this paper is very close to the calculation results of the Nataf transformation method and the orthogonal transformation method. This verifies the effectiveness of the method proposed in this paper. Therefore, the method proposed in this paper can be used to deal with structural reliability problems with correlated variables.

Key words: Rosenblatt transform, dual neural network, Copula function, structural reliability

摘要： 当结构中含有相关性变量时,结构可靠度的求解问题就会变得十分复杂。针对在Rosenblatt变换过程中变量的联合概率密度函数或条件累积分布函数难以求解的问题,提出一种基于Copula函数和对偶神经网络的Rosenblatt变换方法。通过引入Copula函数构造相关性变量的联合概率密度函数。另外,构建对偶神经网络模型,其中一个神经网络学习积分算式中的被积函数部分,另一个神经网络通过与被积函数网络在权值和激活函数上的特定联系,用于构建积分算式中被积函数的原函数,进而实现条件累积分布函数的求解。其中,为提高对偶神经网络的计算效率,分别采用dsigmoid和sigmoid作为被积函数网络和原函数网络的激活函数。该方法打破了Rosenblatt变换在求解结构可靠度时的局限性,拓宽了Rosenblatt变换方法的使用范围。

关键词: Rosenblatt变换, 对偶神经网络, Copula函数, 结构可靠度