## 3 ANN Model
### Input and Output Layer  
As described previously, the strength and the deformation capacities of RC columns are affected by numerous factors. If all of the features are included in the input layer, the ANN model becomes very complex, and prone to fail due to overfitting. Therefore, it is reasonable to select an optimal input feature subset for the ANN model. To achieve optimization, all the input features listed in Table 1, and the output features are first organized into an \(n\times m\) dimensional matrix and an \(n\times 4\) dimensional matrix, respectively, as follows:  
\[\mathbf{x}=\left\{x_{ij}\right\}_{n\times m^{\prime}},\ \mathbf{y}=\left\{y_{jk} \right\}_{n\times 4} \tag{2}\]  
where \(x_{ij}\) (\(i=1,2,\cdots,n\)) is the \(j\)th input feature of the \(i\)th RC column in the training set, and \(y_{jk}\) (\(k=1\), 2, 3, 4) is the corresponding output feature of the \(i\)th RC column. Considering the significant difference in the value ranges of the features, data obtained from the specimens and the test results of the training set are first normalized within the range of [0, 1] by:  
\[\overline{x}_{ij}=\frac{x_{ij}-\min(\mathbf{x}_{j})}{\max(\mathbf{x}_{j})- \min(\mathbf{x}_{j})},\ \overline{y}_{ik}=\frac{y_{jk}-\min(\mathbf{y}_{k})}{\max( \mathbf{y}_{k})-\min(\mathbf{y}_{k})} \tag{3}\]  
where \(\overline{x}_{ij}\) is \(j\)th normalized input feature corresponding to the \(i\)th specimen; \(\mathbf{x}_{j}\) represents the \(j\)th input feature vector of all the specimens; and \(\mathbf{y}_{k}\) denotes the \(k\)th output feature vector. After normalization, \(n\) input-output pairs \(\left\langle\left\{\widetilde{\mathbf{x}_{i}},\widetilde{\mathbf{y}_{i}} \right\},i=1,2,\ldots,n\right\rangle\) are collected into the training set, where \(\widetilde{\mathbf{x}_{i}}=\left\{\overline{x}_{i1},\overline{x}_{i2},\ldots, \overline{x}_{im}\right\}\) and \(\tilde{\mathbf{y}_{i}}=\left\{\overline{y}_{1i},\overline{y}_{2},\ldots, \overline{y}_{ik}\right\}\) are the normalized input and output features of the \(i\)th column, respectively. In this study, a genetic algorithm (GA) was adopted to find the optimal input features of the column. GA is a metaheuristic inspired by the process of theoretical Darwinian natural selection in biological systems to generate high-quality solutions of search problems [33]. In this method, each  
Figure 3: Architecture of the back propagation neural network (BPNN). feature subset, such as structural configuration and material properties, is represented by an individual population. In the selection process, individuals with better phenotypic characteristics have greater probability to survive and reproduce in a population, whereas the less adapted individuals are more likely to disappear. Thus, the GA obtains the optimal solution after a series of iterative computations. Figure 4 shows the chromosome design, fitness function and system architecture for the GA-based optimal feature selection of the columns, which operates in binary spaces (called chromosomes) and manipulates a population of potential solutions for the optimal input subset. A chromosome (genotype of the input features) is represented by binary coding as follows:  
\[\boxed{g_{1},g_{2},\dots,g_{j},\dots,g_{m}}\,\ \ \ g_{j}=\left\{ \begin{array}{ll}1,\ \text{if the $j$-th input feature is selected}\\ 0,\ \text{if the $j$-th input feature is not selected}\end{array} \right. \tag{4}\]  
where \(g_{1},g_{2},\dots,g_{m}\) represent the 24 input features that will be selected. The initial chromosome population is randomly generated first, using binary coding. For example, chromosome 100100 means that the first and the fourth input features were selected. During the selection process, the fitness function is used to evaluate the quality of possible input subsets. There are several measurements can be used to evaluate the accuracy and survival probability of the chromosomes [34].