# Prediction of Compressive Strength Loss of Normal Concrete after Exposure to High Temperature
## 2 Modeling
### Models
#### 2.3.2 PSO-BP Neural Network  
Due to the algorithm of the BP neural network easily resulting in the issue of local extreme small values, the particle swarm algorithm was used in this paper to further optimize weights and thresholds (PSO), developing the PSO-BP neural network. The PSO algorithm is inspired by the predatory behavior of bird colonies, treating each individual as a particle in different spaces to build a search model in terms of velocity (V) and position (X). If there are n particles in an A dimensional space, i.e., X = (X\({}_{1}\), X\({}_{2}\), X\({}_{3}\)...... X\({}_{n}\)), the position of particle X\({}_{i}\) in the A dimensional space, i.e., potential solution, is denoted as X\({}_{i}\) = (X\({}_{i1}\), X\({}_{i2}\), X\({}_{i3}\)...... X\({}_{iA}\))\({}^{\rm T}\). The particle is then updated by each iteration of the individual and extreme global values, with the velocity and position defined in Equations (2) and (3) [90].  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Error (\%)**} & \multicolumn{2}{c}{**6-5-1**} & \multicolumn{2}{c}{**6-7-1**} & \multicolumn{2}{c}{**6-10-1**} & \multicolumn{2}{c}{**6-(5,5)-1**} \\ \cline{2-10}  & **Training** & **Test** & **Training** & **Test** & **Training** & **Test** & **Training** & **Test** \\ \hline RMSE & 6.27 & 6.09 & 4.17 & 3.14 & 5.98 & 4.27 & 4.92 & 3.97 \\ MAE & 5.74 & 5.01 & 3.12 & 2.78 & 4.73 & 3.91 & 4.03 & 3.19 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison between different network structures.  
Figure 2: BP neural network structure.  
\[\mathrm{V_{iA}}^{k+1}=\omega\mathrm{V_{iA}}^{k}+\mathrm{c_{1}r_{1}(P_{iA}-X_{iA}) +c_{2}r_{2}(P_{gA}-X_{iA})}, \tag{2}\]  
\[\mathrm{X_{iA}}^{k+1}=\mathrm{X_{iA}}^{k}+\mathrm{V_{iA}}^{k+1}, \tag{3}\]  
where  
\[\rightarrow \mathrm{V_{i}}\mathrm{:}\]  
particle velocity, denoted as (\(\mathrm{V_{i1}}\), \(\mathrm{V_{i2}}\), \(\mathrm{V_{i3}}\).......\(\mathrm{V_{iA}}\))T;  
\[\rightarrow \mathrm{P_{i}}\mathrm{:}\] individual extreme value, denoted as (\(\mathrm{P_{i1}}\), \(\mathrm{P_{i2}}\), \(\mathrm{P_{i3}}\).......\(\mathrm{P_{iA}}\))T;  
\[\rightarrow \mathrm{P_{g}}\mathrm{:}\] global extreme value, denoted as (\(\mathrm{P_{g1}}\), \(\mathrm{P_{g2}}\), \(\mathrm{P_{g3}}\).......\(\mathrm{P_{gA}}\))T;  
\[\rightarrow \mathrm{X_{i}}\mathrm{:}\] particle position, denoted as (\(\mathrm{X_{i1}}\), \(\mathrm{X_{i2}}\), \(\mathrm{X_{i3}}\).......\(\mathrm{X_{iA}}\))T;  
\[\rightarrow \mathrm{\omega}\mathrm{:}\] inertia weights;  
k: number of current iterations;  
\[\rightarrow \mathrm{c_{1}}\] and \(\mathrm{c_{2}}\): learning factors which are non-negative constants;  
\[\rightarrow \mathrm{r_{1}}\] and \(\mathrm{r_{2}}\): momentum coefficients which are random numbers between [0,1].  
In this paper, particle number n was set as 20, learning factors \(\mathrm{c_{1}}\) and \(\mathrm{c_{2}}\) were kept the same as 2 and momentum coefficients \(\mathrm{r_{1}}\) and \(\mathrm{r_{2}}\) were kept the same as 0.8 [90]. Inertia weight \(\mathrm{\omega}\) was assigned to 0.8 after several trials. In addition, in order to avoid blind searching of particles, the position and velocity of particles were limited in ranges of [\(-\)1, 1] and [0, 1], respectively [90]. The model iterated and updated 100 times, which ensured that the model was sufficiently convergent and could be highly reliable because, after 60 iterations, the adaptability of the model was sufficiently stable, as shown in Figure 3. The training times, network structure, target minimum error, learning efficiency and normalization range of the PSO-BP modeling were kept the same as BP modeling, as shown in the previous section.