## 5 Experiments and Results
### Surrogate Testing  
Testing the surrogate models consisted of calculating the MSE values using the predicted values and the actual fitness values obtained from ARGoS simulations. Table 4 shows the MSE values for each predictor. As we did during the training stage, the configurations producing UAV collisions were removed as they were not taken into account. It can be seen that all GP predictors (except \(gp\_lin\)) performed well, with \(gp\_nn\) being the most promising one, despite Matern being slightly better for swarms of ten robots. As observed during the training process, the proposed ANN predictors did not produce results good enough to compete with GPs.  
We also studied the time elapsed for calculating the fitness for a given configuration of a UAV swarm using the different surrogate models and compared them to simulations using ARGoS. Table 5 shows the different average execution times calculated from 30 evaluations per swarm. We can see that all the surrogate models are faster than ARGoS simulations, as expected. Moreover, the elapsed prediction times showed minimal variations with respect to the number of UAVs. This is especially interesting for 30 UAVs, where the achieved increase in speed was more than 3600.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**\# UAVs** & **\# Obs.** & **gp\_lin** & **gp\_sexp** & **gp\_nn** & **gp\_m32** & **gp\_m52** & **ann** \\ \hline
3 & 2220 & 12.564 & 6.596 & **6.199** & 6.279 & 6.399 & 8.943 \\
5 & 1853 & 13.483 & 5.471 & **5.146** & 5.231 & 5.314 & 6.938 \\
10 & 1096 & 11.680 & 3.977 & 3.966 & **3.841** & 3.872 & 5.319 \\
15 & 923 & 15.797 & 4.635 & **4.349** & 4.455 & 4.563 & 4.879 \\
30 & 728 & 20.714 & 6.258 & **6.004** & 6.115 & 6.250 & 8.622 \\ \hline \hline \end{tabular}
\end{table}
Table 4: MSE values for the predictions performed using the GP and ANN models compared with ARGoS simulations. Best values are in bold.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**\# UAVs** & **\# Obs.** & **gp\_lin** & **gp\_sexp** & **gp\_nn** & **gp\_m32** & **gp\_m52** & **ann** \\ \hline
3 & 238 & 13.030 & 6.138 & **5.081** & 5.408 & 5.835 & 10.010 \\
5 & 192 & * 11.665 & 4.059 & **3.643** & 3.720 & 3.898 & 5.295 \\
10 & 118 & 7.542 & 2.236 & **0.995** & 1.759 & 1.926 & 2.525 \\
15 & 119 & 12.042 & 5.083 & **4.486** & 4.747 & 4.980 & 4.398 \\
30 & 86 & 16.882 & 6.224 & **5.735** & 5.926 & 6.187 & 6.220 \\ \hline \hline \end{tabular}
\end{table}
Table 2: MSE values for the training process of predictors based on GPs and an ANN. Note that \(gp\_lin\) (*) did not converge for five UAVs. Best values are in bold.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**\# UAVs** & **\# Obs.** & **gp\_lin** & **gp\_sexp** & **gp\_nn** & **gp\_m32** & **gp\_m52** & **ann** \\ \hline
3 & 238 & **0.457** & 1.211 & 2.012 & 1.021 & 1.128 & 45.934 \\
5 & 192 & * 6.224 & **0.558** & 0.769 & 0.938 & 0.940 & 29.205 \\
10 & 118 & **0.093** & 0.198 & 0.406 & 0.362 & 0.345 & 9.399 \\
15 & 119 & **0.195** & 0.337 & 0.396 & 0.327 & 0.327 & 2.077 \\
30 & 86 & **0.122** & 0.294 & 0.182 & 0.260 & 0.220 & 0.772 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Elapsed training times in seconds for the six predictors. Note that \(gp\_lin\) (*) did not converge for five UAVs. Best values are in bold.  
In the following section, we use the surrogate model based on \(gp\_nn\) to predict the fitness value of the different configurations calculated during the optimisation of the UAV swarm using our proposed HEA.