# Improving Particle Swarm Optimization Analysis Using Differential Models
## 4 Differential Evolution PSO
### Instance Simulation  
For the performance comparison of typical test functions such as the F\({}_{8}\) function and F\({}_{10}\)-F\({}_{13}\) function, please refer to Figure 2. The abscissa of Figure 2 represents the evolutionary algebra, and the ordinate represents the average optimal fitness value. The differential PSO algorithm with Runge-Kutta method (DPSO-RK) is better than the differential PSO algorithm with Euler (DPSO-E) and the differential PSO with modified Euler( DPSO-ME). This is very easy to understand numerically since the 4 th-order Runge-Kutta method has a 4 th-order local truncation error, so it can more accurately track the solution of the differential model (26).  
To further verify the performance of the algorithm DPSO-RK, three other algorithms were selected: standard PSO (SPSO) [27; 28], modified time-varying accelerator coefficient PSO (MPSO-TVAC) [29; 30], and comprehensive learning PSO (CLPSO) [31; 32]. When the dimension of the test function is high (dimension is 100-300), the differential evolution PSO based on the Runge-Kutta method works very well, far better than the other three algorithms, as shown in Figure 3. The absolute stability selection of the step size makes the algorithm have strong global searchability to better jump out of the local extreme point. Tables 2 and 3 present execution time of different PSO algorithms in F8-F13 (test function). The CPU of our computer running the program is Intel(R) Core (TM) i5-9400F, 2.90 GHz. The operating system is Microsoft Windows 11. The memory is 32 GB and the MATLAB version is 2021b.  
Figure 2: \(Cont.\)  
\begin{table}
\begin{tabular}{l l l} \hline
**Algorithm Test Item** & \multicolumn{2}{c}{**Execution Time (Second)**} \\ \hline \multirow{3}{*}{\(\text{F}_{8}\) (Schwefel function)} & DPSO-E & 42.56 \\ \cline{2-3}  & DPSO-ME & 50.23 \\ \cline{2-3}  & DPSO-RK & 72.45 \\ \hline \multirow{3}{*}{\(\text{F}_{10}\) (Ackley function)} & DPSO-E & 63.38 \\ \cline{2-3}  & DPSO-ME & 68.41 \\ \cline{2-3}  & DPSO-RK & 74.32 \\ \hline \multirow{3}{*}{\(\text{F}_{11}\) (Griewank function)} & DPSO-E & 81.45 \\ \cline{2-3}  & DPSO-ME & 85.21 \\ \cline{2-3}  & DPSO-RK & 90.42 \\ \hline \multirow{3}{*}{\(\text{F}_{12}\) (Penalized function)} & DPSO-E & 106.23 \\ \cline{2-3}  & DPSO-ME & 108.32 \\ \cline{2-3}  & DPSO-RK & 110.45 \\ \cline{2-3}  & DPSO-E & 124.26 \\ \cline{2-3}  & DPSO-ME & 126.23 \\ \cline{2-3}  & DPSO-RK & 128.43 \\ \hline \end{tabular}
\end{table}
Table 2: Algorithm execution time of 3 DPSO in \(\text{F}_{8}\)–\(\text{F}_{13}\).  
Figure 2: Comparison of dynamic performance of three DPSO algorithms in the case of 300 dimensions. (**a**) \(\text{F}_{8}\) function, (**b**) \(\text{F}_{10}\) function, (**c**) \(\text{F}_{11}\) function, (**d**) \(\text{F}_{12}\) function, and (**e**) \(\text{F}_{13}\) function, (**f**) algorithm icon illustration.  
Figure 3: Comparison of dynamic performance of four algorithms in the case of 300 dimensions. (**a**) F8 function, (**b**) F10 function, (**c**) F11 function, (**d**) F12 function, and (**e**) F13 function. (**f**) algorithm icon illustration.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Algorithm Test Item** & \multicolumn{2}{c}{**Execution Time (Second)**} \\ \hline  & SPSO & 43.15 \\ \cline{2-3} F8 (Schwefel function) & MPSO-TVAC & 45.46 \\ \cline{2-3}  & CLPSO & 47.21 \\ \cline{2-3}  & DPSO-RK & 48.51 \\ \hline  & SPSO & 65.22 \\ \cline{2-3} F10 (Ackley function) & MPSO-TVAC & 68.41 \\ \cline{2-3}  & CLPSO & 71.35 \\ \cline{2-3}  & DPSO-RK & 73.24 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The algorithm execution time of different 4 PSOs in F8–F13.