## 2 Material and Methods
### Particle Swarm Optimization Algorithm  
Particle Swarm Optimization (PSO) is a population-based metaheuristic optimization algorithm inspired by the social behavior of birds flocking or fish schooling. [PERSON] and [PERSON] first proposed the algorithm in 1995 [39]. The main aim of the PSO algorithm is to maintain a group of particles in a search space, where each particle represents a potential solution. A position vector and velocity vector characterize each particle. The position vector represents the potential solution in the search space, and the velocity vector determines the direction and speed of the particle's movement in the search space [40,41]. The best-known position of the swarms has been widely used to solve various optimization problems in recent years due to its simplicity, effectiveness, and ability to handle non-linear, non-convex, and multi-modal problems [42]. Recent research in PSO has focused on improving  
Figure 1: The number of hidden layer nodes. its performance, scalability, and applicability to various optimization problems. Hybrid algorithms combine the strengths of multiple algorithms to overcome their limitations. For example, PSO has been combined with the genetic algorithm (GA) to form a hybrid algorithm called PSO-GA [43], which has been shown to improve the performance of both algorithms. Similarly, PSO has been hybridized with Differential Evolution (DE) [44], which has been applied to solve complex optimization problems. [45] proposed an improved PSO algorithm that used a novel local search strategy to enhance its search performance for large-scale optimization. The proposed algorithm employs a guide search strategy that combines PSO with a local search method to accelerate the convergence and improve the diversity of the swarm. Ref. [46] presented a PSO algorithm that used a surrogate-assisted framework to solve multi-objective optimization. The proposed algorithm employs a surrogate model to approximate the objective functions, which can significantly reduce the computational cost and enhance the search performance. PSO has also been extended to solve multi-objective optimization problems using multi-objective PSO algorithms. Recent work has focused on improving the performance of multi-objective PSO algorithms by incorporating adaptive strategies, using Pareto dominance, and combining multi-objective PSO with other optimization algorithms [47]. Moreover, parallel PSO algorithms have been developed to take advantage of modern parallel computing architectures. One approach is to use parallel evaluation to speed up the evaluation of fitness functions [48]. Another is to use parallel population evaluation to accelerate the selection process. The PSO algorithm starts by initializing a population of particles randomly in the search space. Each particle evaluates its fitness value based on the objective function to be optimized. A fitness function is defined to evaluate the fitness of each particle based on its position in the search space. The fitness function measures how well the particle's position solves the optimization problem and is used to determine the quality of the candidate solutions [49]. The particles update their position and velocity vectors based on their own best solution (personal best) and the best solution found by any particle in the swarm (global best). The velocity vector determines the direction and speed of the particle's movement, while the position vector determines the particle's new position in the search space [50]. In PSO, a swarm of particles is used to explore a search space in search of an optimal solution to an optimization problem. Each particle in the swarm represents a candidate solution to the problem and is located in a D-dimensional search space, where D is the number of problem variables [51]. The particles are initialized randomly within the search space, and their positions and velocities are updated iteratively based on their own best position and the best position of their neighbors in the swarm [50].