## 3 Reliability Analysis Method Based on PSO-BPNN
### PSO-BPNN Method  
PSO is a population-based optimization algorithm inspired by the social and behavioral characteristics of biological systems. It is designed to solve complex optimization problems by iteratively improving candidate solutions based on a predefined fitness function. PSO has significant advantages, such as its ability to escape local extrema and its strong capability for finding global optima [30]. In the PSO algorithm, the objective function is referred to as the fitness function, which evaluates the quality of each candidate solution. Each particle within the search space is characterized by its position and velocity, and its fitness value is determined by the fitness function. The particles iteratively update their positions and velocities by comparing their personal best fitness value (individual best) with the global best fitness value (shared among the swarm). Through this process of interaction and adjustment, the particle population converges to the optimal value of the fitness function, thereby achieving optimization of the objective function [31]. The predefined fitness function in this study is shown in Equation (9). \[\text{fit}(x)=\frac{1}{c+1+Y+my} \tag{9}\]  
where \(c\) is the estimated value of the limit state function \(Y=g(\textbf{X});y\) is the penalty term, representing the degree of constraint violation; \(m\) is the penalty coefficient, controlling the weight of the penalty term. The fitness function was designed to fulfill two main objectives: (1) minimize the unreliability by reducing the value of \(c\), and (2) penalize solutions that violate physical or operational constraints. The term \(\frac{1}{c+1}\) ensures that particles closer to the failure threshold receive lower fitness values, encouraging the search toward safer and more reliable solutions. Meanwhile, the penalty term \(Y\) and its coefficient \(m\) are crucial for guiding the optimization process toward feasible regions of the design space. The value of \(m\) was determined through a sensitivity analysis to balance constraint satisfaction and algorithm convergence, ensuring that the penalty neither overwhelms the optimization process nor allows infeasible solutions to persist. The choice of this fitness function is crucial for achieving effective optimization within the PSO-BPNN framework. As the PSO algorithm iteratively adjusts the particles' positions and velocities, the fitness function plays a central role in evaluating the quality of each candidate solution. By minimizing the fitness value, the PSO algorithm optimizes the neural network's parameters, improving its ability to approximate the implicit limit state function and predict failure probabilities accurately. The PSO-BPNN method integrates PSO with the BPNN by treating the weights and biases of the neural network as particles within the swarm. PSO is employed to optimize these particles by iteratively updating their positions (the weights and biases) and velocities according to a predefined fitness function. The fitness function evaluates the performance of the neural network, aiming to minimize the error of the neural network. Through this process, the PSO-BPNN method seeks to identify the optimal set of weights and biases that enhance the accuracy and generalization capability of the BPNN. This ultimately establishes a more precise input-output relationship, thus improving the network's overall predictive performance. In the standard PSO algorithm, the particle swarm is composed of \(\mathbf{N}=(N_{1},N_{2},N_{3},\ldots,N_{n})\) particles. The position of the \(i\)th particle is represented by \(\mathbf{x}_{i}=(x_{1i},x_{i2},x_{i3},\ldots,x_{id})\), and its velocity is denoted as \(\mathbf{v}_{i}=(v_{i1},v_{i2},v_{i3},\ldots,v_{id})\). At the \(i\)th iteration, the velocity and position of the particle are updated for the \((t+1)\)th iteration according to Equations (10) and (11).