# Developing a Library of Shear Walls Database and the Neural Network Based Predictive Meta-Model
## Modeling the Network
### Performance of the Network  
Once the network is trained, it can be used to define the complex relationship among the input parameters, and to predict the output for any new SW model. Generalization means estimating the value on the hyper-surface where there are no available data. Mathematically, the learning process is a nonlinear curve-fitting algorithm, while generalization is the interpolation and extrapolation of the input data [47]. The ability of a network to generate new outputs depends on the number of parameters involved in the problem, the complexity of the parameters, and the structure of the network.  
The neural network is constructed based on the number of neurons obtained from previous section. Figure 3 shows the architecture of the proposed networks for both SWs. In this paper, two MLPs are used to estimate the load bearing capacity, stiffness, and drift ratio of the SPSWs and RCSWs.  
Overall, 70% of data were used for training, 15% for validation, and 15% for testing. The validation and test data monitor the network over-training and performance, respectively. Performance of the network is shown in Figure 4. It states that the network has specific MSE for \(n\)th epoch, and there is no un-convergence or over-fitting.  
Figure 4: _Cont._  
Figure 3: Proposed ANN models for shear walls. (**a**) SPSW, (**b**) RCSW.  
Furthermore, Figure 5 illustrates the quality of the estimation as a function of coefficient of determination, R2, in both SPSW and RCSW models. Finally, the MSE values for these data are summarized in Table 5. The proper performance of the network is evident in the estimation of the strength, stiffness and drift ratio of SWs. Therefore, the proposed network can learn the relation between the input and output parameters, and provide the results with appropriate accuracy. Finally, the true values of the strength, stiffness and drift ratio from experimental data and the predictive ANN-based meta-model are compared in Figure 6.  
Figure 4: Performance function of the proposed network. (**a**) SPSW; Stiffness, (**b**) SPSW; Strength, (**c**) RCSW; Strength, (**d**) RCSW; Stiffness, (**e**) RCSW; Drift.  
Figure 5: _Cont._  
Figure 5: Regression values for the proposed ANN meta-model. (**a**) SPSW; Strength, (**b**) SPSW; Stiffness, (**c**) RCSW; Strength, (**d**) RCSW; Stiffness, (**e**) RCSW; Drift ratio.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Output** & **Train** & **Validation** & **Test** \\ \hline Strength of SPSW & 2.49 \(\times 10^{-4}\) & 0.0046 & 2.28 \(\times 10^{-4}\) \\ Stiffness of SPSW & 1.64 \(\times 10^{-5}\) & 0.0043 & 0.0108 \\ Strength of RCSW & 0.00743 & 0.0034 & 0.00123 \\ Stiffness of RCSW & 6.45 \(\times 10^{-4}\) & 0.0038 & 0.0061 \\ Drift ratio of RCSW & 0.0069 & 0.0059 & 0.0056 \\ \hline \hline \end{tabular}
\end{table}
Table 5: MSE values for training, validation and test data.  
Thus far, a detailed performance is discussed with respect to MSE. However, one may evaluate the predictive meta-models using other statistical indicators such as root mean square error (RMSE), Nash-Sutcliffe efficiency (NSE) coefficient, mean absolute error (MAE), and correlation coefficient (R). These metrics can be computed using Equation (2).  
\[\begin{split}\text{RMSE}=\sqrt{\frac{\sum(\hat{y}-y)^{2}}{N}}& \text{NSE}=1-\frac{\sum(\hat{y}-y)^{2}}{\sum(y-\hat{y})^{2}}\\ \text{MAE}=\frac{100}{N}\sum\left|\frac{y-\tilde{y}}{y}\right|& \text{R}=\frac{\sum(\hat{y}-\tilde{y})(y-\tilde{y})}{\sqrt{\sum(\hat{y}- \tilde{y})^{2}}\sqrt{\sum(y-\tilde{y})^{2}}}\end{split} \tag{2}\]  
Table 6 compares these metrics (including MSE) based on all data points obtained from the networks. As seen, there is a good consistency among those five metrics for five meta-models. For example, higher R corresponds with lower RMSE.  
Figure 6: Comparison of measured experiments and predicted ANN meta-model; Note: sorted in ascending order based on experimental data. (**a**) SPSW; Strength, (**b**) SPSW; Stiffness, (**c**) RCSW; Strength, (**d**) RCSW; Stiffness, (**e**) RCSW; Drift.