## 5 Prediction Models  
### Artificial Neural Network (ANN)  
Artificial neural networks (ANNs) are computational models inspired by the structure and function of biological neural networks. ANNs excel at modeling non-linear relationships between inputs and outputs, making them a valuable tool for predicting complex phenomena in concrete material science. In this study, an artificial neural network featuring a forward multi-layered perceptron structure was utilized. The backpropagation training algorithm employed the Levenberg-Marquardt (LM) algorithm optimization method, along with the scaled conjugate gradient (SCG) algorithm. The LM algorithm partitioned input and output vectors into training, validation, and test datasets. The SCG algorithm, by eliminating the need for second derivative calculations, ensured convergence to the quadratic function minimum, thus bypassing the time-intensive linear search process. The study's input variables included cement, silica fume, water, and naphthalene-based superplasticizer. Table 7 presents a report on the characteristics of these input and output variables. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Independent variables (kg/m\({}^{3}\))** & & & & \\ \hline  & **Min.** & **Max.** & **Averg.** & **SD.** \\ \hline  & 640 & 870 & 763 & 75.3 \\ \hline  & Silica fume (SF) & 104 & 200 & 137.24 & 26.58 \\ \hline  & Water (W) & 144 & 220 & 180 & 19.44 \\ \hline  & 6 & 9 & 7.2 & 0.832 \\ \hline  & & & & \\ \hline  & **Min** & **Max** & **Averg.** & **SD.** \\ \hline  & 52.08 & 81.60 & 70.76 & 29.1 \\ \hline  & Compressive strength—28 days & 115.07 & 145.44 & 132.05 & 9.13 \\ \hline  & & & & \\ \end{tabular}
\end{table} 
Table 7: Characteristics of the independent and dependent variables. Figure 4: The squeezing test: (**a**) filling the ring mold, (**b**) removing the mold, (**c**) squeezing a high-viscosity mixture, and (**d**) squeezing a low-viscosity mixture. To determine the optimal number of layers and neurons in the hidden layer of an artificial neural network, the learning and error method was utilized. The stop criterion for network training was selected as the mean squared error (MSE) (Equation (1)), representing the mean squared difference between the model's predicted value (network output) and the actual target value. The R\({}^{2}\) value indicates the extent and strength of the relationship and dependency between the network's output value and the target, always ranging between 1 and \(-1\) (Equation (2)). The closer the value is to 1, the stronger the relationship between the network's output and the target value. In this study, 10 neurons were chosen in the hidden layer to forecast the compressive strength at 7 and 28 days. Thus, the neural network's configuration in this study is 4-10-2: 4 inputs, 10 neurons in the hidden layer, and 2 outputs. Figure 5 depicts the overall architecture of the artificial neural network modeled in this study. \[MSE=\frac{\sum_{j=0}^{P}\sum_{i=0}^{N}\bigl{(}d_{ij}-y_{ij}\bigr{)}^{2}}{N\times P}, \tag{1}\]  
\[R^{2}=1-\frac{\sum_{i=1}^{N}(y_{i}-\hat{y}_{i})^{2}}{\sum_{i=1}^{N}(y_{i}-\overline {y}_{i})^{2}}, \tag{2}\]  
In Equations (1) and (2), the parameters are defined as follows: MSE (mean squared error) measures the average squared difference between the predicted values (\(y_{ij}\)) and the actual observed values (\(d_{ij}\)) across all samples, where N represents the total number of samples and P denotes the number of predictors or observations per sample. On the other hand, \(R^{2}\) (coefficient of determination) evaluates the proportion of variance in the dependent variable (\(y_{i}\)) that is predictable from the independent variables. In the numerator, \(\sum(y_{i}-\hat{y}_{i})^{2}\) captures the sum of squared residuals, while the denominator \(\sum(y_{i}-\hat{y}_{i})^{2}\) accounts for the total variance, with \(\hat{y}_{i}\) representing predicted values and \(\overline{y}_{i}\) denoting the mean of observed values.