# Improving Particle Swarm Optimization Analysis Using Differential Models
## 1 Introduction  
Since the introduction of PSO, many scholars have improved its performance from different viewpoints. Scientists have proposed inertia weights, contraction factors, PSOs with time acceleration constants, and understanding learning and random PSOs from a mathematical point of view. Researchers have proposed PSO with escape behavior from the viewpoint of biology and also proposed PSO with Kalman filtering and a Bayesian network PSO from the viewpoint of information theory [1, 2]. Many scholars have proposed quantum PSO from the field of physics. Therefore, researchers are faced with many improved algorithms and methods to analyze and utilize their advantages. The above process can also further improve the performance of the algorithm. However, processing and analyzing these algorithms will become a more difficult problem [3, 4]. In 2006, Professor [PERSON] pointed out that the field of intelligent computing needs to establish a unified model to effectively analyze the various intelligent algorithms that have been proposed; this is a developing research field [5]. The establishment of a unified model of smart algorithms requires the effective integration of each algorithm itself, establishing a unified model for all smart algorithms. Because the PSO algorithm was proposed late, various theoretical systems have not yet been established, so it is urgent to establish such aunified model [6; 7; 8]. Many researchers believe that the unified intelligent algorithm model is in urgent need of research. Thus, a preliminary unified model of PSO is given from the perspective of software implementation. Differential equations refer to equations that describe the relationship between the derivatives and arguments of unknown functions; difference equations, also known as recurrence relations, are equations that contain unknown functions and their differences, but do not have derivatives. The solution of a differential equation is a function that satisfies the equation--that is, an analytical solution. In the algebraic equation of elementary mathematics, its solution is a constant value [9; 10]. Usually, the differential equation model is more suitable for solving continuous problems, which is its advantage. The differential equation model is less suitable for dealing with discrete problems, which is the disadvantage of the differential equation model. The difference equation replaces the description of the differential equation, and the derivative function is avoided in the equation, which can be solved by iterative operation. Difference equations are more suitable for discrete problems. However, the accuracy of the difference equation model is slightly lower (the secant line is used instead of the tangent line), which is a disadvantage of the difference equation model. Please refer to the description in Figure 1.  
Difference equations reflect the values and changing laws of discrete variables. The difference equation can be established using the equilibrium relationship satisfied by the values of one or several discrete variables. Differential equations reflect the values and changing laws of continuous variables [11; 12]. The micro-element method means that when building a dynamic model of an object that changes continuously in time or space, it often considers small unit changes in time or space because the balanced relationship between these micro-elements is relatively simple and easy to use. The method of differential calculus is processed. Such models are given in the form of differential equations [13; 14]. In the recent PSO-related research literature, some excellent papers have been published [15; 16]. For example, [PERSON] et al. proposed the use of PSO technology to optimize the design of this supported viscous damper and pendulum-tuned mass damper [17]. Authors such as [PERSON] published a paper on a multi-objective PSO algorithm based on a two-archive mechanism [18]. [PERSON] et al. used the PSO algorithm to practice LSTM neural networks for rainfall-runoff simulation [19].