## 7 Result and Equation Extraction from Model  
The performance of the developed neural network model for fatigue crack area prediction was evaluated using training, validation, and testing datasets, as illustrated in Figure 9. The scatter plots compare the predicted fatigue crack area (m\({}^{2}\)) with the measured fatigue crack area (m\({}^{2}\)), assessing the model's predictive accuracy.  
In Figure 9a, the training dataset exhibits a robust correlation (R\({}^{2}\) = 0.91), accompanied by relatively minimal errors (RMSE = 2.43 m\({}^{2}\), MAE = 1.86 m\({}^{2}\)), suggesting that the model proficiently comprehends the underlying data patterns. Figure 9b delineates the testing dataset, which attains the highest correlation (R\({}^{2}\) = 0.93), albeit with a marginal increase in RMSE (3.44 m\({}^{2}\)) and MAE (2.67 m\({}^{2}\)), implying a degree of variability in the predictions. Figure 9c portrays the results from the validation dataset, wherein the model sustains significant predictive accuracy with R\({}^{2}\) = 0.88, an RMSE of 2.42 m\({}^{2}\), and an MAE of 1.71 m\({}^{2}\). Finally, Figure 9d amalgamates all the datasets, producing an overall R\({}^{2}\) = 0.90, an RMSE of 1.83 m\({}^{2}\), and an MAE of 0.89 m\({}^{2}\), thereby affirming the model's dependability and capacity for generalization.  
As mentioned earlier, this study also aims to develop an equation from the ANN model that can estimate the Fatigue Cracking Area in the pavement without needing to run the full model. In Equation (3), the hyperbolic tangent (tanh) activation function plays a key role in shaping predictions. The tanh function helps the model capture complex relationships between different pavement factors, making it more adaptable to real-world  
Figure 8: Schematic diagram of the final ANN model used in this study.  
conditions. Since tanh scales value between \(-1\) and \(1\), it ensures that the input variables contribute in a balanced way, preventing extreme values from dominating the prediction. This also helps stabilize the learning process and improves the model's overall accuracy. By using tanh in this equation, the model effectively captures nonlinear patterns in pavement behavior, leading to more accurate and reliable fatigue cracking predictions. The equation below represents the predictive model.  
\[\begin{split}\text{FCA = 24.6 \times(-0.739 \times tanh (-3.931 \times TH-6.853 \times AV+7.564 \times AC+0.032 \times KE-0.521 \times TEMP-499.075}\\ \times\text{CAR + 20.035) + 0.815 \times tanh (0.252 \times TH-2.473 \times AV+2.524 \times AC+0.010 \times KE-0.408 \times TEMP-551.392}\\ \times\text{CAR + 9.556) - 0.776 + 1) + 0.200}\end{split} \tag{3}\]  
Here  
FCA = Fatigue Cracking Area (m\({}^{2}\));  
TH = Thickness of HMA layer (in);  
AV = Air Void (%);  
AC = Asphalt Content (%);  
KE = Cumulative ESAL (KESAL);  
TEMP = Average Annual Temperature (\({}^{\circ}\)C);  
CAR = Normalized Comprehensive Area Ratio (CAr\({}^{\prime}\)).  
Figure 9: ANN model correlation coefficient (R\({}^{2}\)), (**a**) training dataset, (**b**) validation dataset, (**c**) testing dataset, and (**d**) all datasets.  
The input weight matrix (W\({}_{\text{ih}}\)), input bias matrix (b\({}_{\text{ih}}\)), output weight matrix (W\({}^{\prime}\)\({}_{\text{ho}}\)), and output bias matrix (b\({}_{\text{ho}}\)) that govern the ANN model are shown below in Equations (4)-(7):  
\[\text{Wih}=\begin{bmatrix}-13.1681&-15.1786&8.6984&7.2567&-3.2583&-4.6381\\ 0.8441&-5.4767&2.9022&2.2401&-2.5529&-5.1243\end{bmatrix} \tag{4}\]  
\[\text{b}_{\text{ih}}=\begin{bmatrix}1.500\\ 3.149\end{bmatrix} \tag{5}\]  
\[\text{W}^{{}^{\prime}}_{\text{ho}}=\begin{bmatrix}-0.739\\ 0.815\end{bmatrix} \tag{6}\]  
\[\text{b}_{\text{ho}}=[-0.776] \tag{7}\]