## 3 Reliability Analysis Method Based on PSO-BPNN
### Failure Probability Calculation Process Based on the PSO-BPNN Method  
This paper proposes a method for structural reliability analysis that integrates the particle swarm optimization back propagation neural network to calculate failure probability. By employing the PSO-BPNN method, it addresses the limitations of the traditional MCS, such as high computational cost and slow convergence rate. The core concept of the traditional MCS method is to approximate the population mean using the sample mean when the sample size is sufficiently large. The fitted limit state function \(\overline{g}(\mathbf{X})\) divides the input variable space into the failure domain \(F=\{\mathbf{x}|g(\mathbf{x})\leq 0\}\) and the safe domain \(\mathsf{S}=\{\mathbf{x}|g(\mathbf{x})>0\}\). To distinguish between failure and safe states in structural reliability analysis, a failure domain indicator function \(I_{F}\big{(}\mathbf{x}_{j}\big{)}\) is introduced. If \(\overline{g}(\mathbf{  
The MCS method is straightforward to implement, and its accuracy converges to the true solution as the sample size increases. However, in practical engineering applications, the structural limit state function is often highly complex or even implicit, such as a finite element model. Evaluating the limit state function for each sample to compute response values involves considerable computational cost. To address this issue, the method proposed in this paper is designed to overcome the computational challenges associated with the MCS method in structural reliability analysis. By introducing a substitution of limit state function, the computational burden of repeatedly evaluating the limit state function can be significantly reduced. Specifically, a neural network is trained to approximate the functional relationship of the limit state function, thereby eliminating the need to directly evaluate the original function. The detailed steps are as follows:  
1. Collection and preprocessing of sample data. In this step, the Latin Hypercube Sampling (LHS) method is used to generate \(K\) sets of input data by sampling the random variables \(\mathbf{X}=\{X_{1},X_{2},\ldots,X_{n}\}\). For each input sample, the performance response is calculated, and the corresponding limit state function \(g(\mathbf{X})\) is evaluated to obtain \(K\) sets of output results. To ensure numerical stability and prevent data with small magnitudes from being annihilated during computation, it is essential to normalize all input and output parameters. Since the activation function used between the input layer and the hidden layer is an S-shaped function, the data are normalized and mapped to the [0, 1] interval. This ensures compatibility with the activation function and facilitates better con-vergence during training. The mapping function for normalization is provided in Equation (14): \[x_{\text{st}}=\frac{x-x_{\text{min}}}{x_{\text{max}}-x_{\text{min}}}\] (14) where \(x_{\text{st}}\) represents the normalized data; \(x\) denotes the original data; \(x_{\text{max}}\) and \(x_{\text{min}}\) correspond to the maximum and minimum values of the original dataset for each parameter, respectively. 2. Determine the structure of the BPNN. The structure of BPNN is defined employing \(\mathbf{X}=\{X_{1},X_{2},\ldots,X_{n}\}\) as the system input and \(g(\mathbf{X})\) as the output variable. 3. Split dataset. Divide the dataset into a training set and a testing set, where 80% of the sample data are allocated to the training set, while the remaining 20% are used as the testing set. This division is a critical step in effectively training the BPNN, as it helps mitigate the risk of overfitting and ensures the model exhibits strong generalization capability. 4. Configurate training parameters. Prior to training the BPNN, the training parameters must be appropriately configured. Specifically, the learning rate is set to 0.01, the maximum number of training iterations is set to 1000, and other parameters are maintained at their default values.