## 4 PSO Based on the PSO-BP Model  
### PSO-BP Description  
The particle swarm optimization (PSO) algorithm is a metaheuristic approach grounded in swarm intelligence, drawing inspiration from the collective foraging behaviors of biological populations--such as bird flocks or fish schools--in complex environments [28]. By mimicking the decentralized decision-making and dynamic coordination observed in these natural systems, PSO iteratively refines candidate solutions through social interaction and individual experience. This synergy enables efficient exploration of high-dimensional solution spaces, making it particularly effective for optimizing nonlinear, multimodal problems in fields ranging from agricultural engineering to materials science. In this algorithm, each candidate solution is abstracted as a particle with multidimensional attributes. These particles collectively achieve global optimization through iterative information exchange between individual experiences and swarm intelligence [29]. Specifically, each particle in the solution space possesses dynamically updated position and velocity vectors: The position vector (xi) represents the current solution's quality, encoding parameter values (e.g., friction coefficients, surface energy); The velocity vector (vi) dictates the search direction and step size, balancing exploration (global search) and exploitation (local refinement).  
During the iterative process, particles adhere to three core rules:1. Individual Experience Memory Mechanism: Retention of the particle's historical optimal position (p\({}_{\text{best}}\)).
2. Swarm Information Sharing Mechanism: Tracking of the swarm's global optimal position (g\({}_{\text{best}}\)).
3. Motion Inertia Preservation Mechanism: Balancing exploration (global search) and exploitation (local refinement) through velocity damping [30].  
Compared to traditional backpropagation (BP) neural networks, the PSO-BP hybrid model achieves synergistic optimization of network parameters through the integration of swarm intelligence optimization mechanisms. Traditional BP networks rely on gradient descent for parameter adjustment, which is prone to converging on local optima and exhibits high sensitivity to initial values [31]. In contrast, the PSO-BP model maps neural network weights and thresholds into the multidimensional solution space of a particle swarm, enabling global exploration of the parameter space through collaborative search mechanisms. This novel hybrid architecture endows the network with three key optimization characteristics:  
* Adaptive Parameter Adjustment: Reduces dependency on initial values by dynamically tuning learning rates and momentum terms.
* Balanced Search Strategy: Maintains equilibrium between local refinement (exploitation) and global exploration via velocity-controlled particle dynamics.
* Enhanced Robustness and Generalization: Improves tolerance to noisy or incomplete datasets while minimizing overfitting risks.  
According to the study of [32], the hybrid model significantly outperforms traditional methods in metrics such as convergence speed and classification accuracy, with particularly prominent advantages when handling nonlinear complex systems.