# Predictive Modeling and Experimental Analysis of Cyclic Shear Behavior in Sand-Fly Ash Mixtures
## 5 Data Analysis
The training error (blue line) decreases rapidly in the initial epochs, which indicates that the model quickly learns the primary stress response patterns from the sand and sand-fly ash mixture data. After reaching its best performance with an MSE of 1.7321 at epoch 184, the training error stabilizes, showing that the model has effectively minimized error without further improvement. Meanwhile, the testing error (red line) remains higher than the training error, suggesting that although the model generalizes well, it encounters some variance when predicting unseen stress values in the final loading cycle. The constant error level after epoch 184 also indicates that the model avoided over fitting, as the testing error does not diverge or show significant fluctuations. Figure 12 provides details on various parameters tracked during training, such as the gradient, mu, number of parameters, and sum-squared parameters. The gradient decreases  
Figure 11: Best training performance of NARX model. steadily and ultimately reaches a low value (0.12488), signaling that the model's error rate changes minimally with respect to parameter adjustments, indicating convergence. The mu parameter, which represents the adaptive regularization term for Bayesian regularization, has reached a high value (\(5\times 10^{10}\)) by the end of training, indicating that regularization was heavily applied to balance fitting accuracy with generalizability. The number of parameters and sum-squared parameters remain relatively constant throughout the training process, reinforcing the stability of the model. Additionally, the validation checks remained at zero throughout the training, suggesting that the model did not require early stopping due to over fitting. Figure 13 displays the response of the model's output against the target values for a loading series, showing how accurately the model captures cyclic behavior. The output response (in blue) aligns closely with the target values (in red) across several cycles, confirming that the model successfully learned the cyclic patterns. Minor discrepancies are visible, especially toward the peaks and troughs of the cycles, where slight deviations suggest that the model occasionally underestimates or overestimates extreme values. However, the error plot at the bottom of Figure 13 shows minimal error throughout the series, with only a few noticeable spikes, indicating high prediction accuracy and minimal residual error. The performance plots suggest that the NARX neural network model is highly effective in capturing the cyclic behavior of the fly ash-sand mixtures with a well-minimized error. It adequately generalizes unseen data, as indicated by the stable test error, though some minor deviations may still exist at extreme points in the cyclic response. Figure 14 illustrates the comparison between the measured and predicted horizontal stress-strain responses for clean sand (TS) and sand-fly ash mixtures containing varying percentages of fly ash (2.5%, 5%, 10%, 15%, and 20%, denoted as FA2.5, FA5, FA10, FA15, and FA20). These results correspond to the fifth and final cycle of constant load cyclic direct shear testing. The predicted responses were obtained using a NARX model, which aims to replicate the cyclic behavior of these soil mixtures under dynamic loading conditions. The plots demonstrate a close alignment between the measured and predicted data for all tested specimens, suggesting that the NARX model performs effectively in capturing  
Figure 12: Training state of NARX model. the cyclic stress-strain behavior. For clean sand (TS), the stress-strain response exhibits a typical hysteresis loop, and the model accurately reflects the cyclic loading pattern. In low fly ash mixtures, such as FA2.5 and FA5, the predicted and measured results show only minor discrepancies, with the model effectively accounting for the slight modifications in soil behavior introduced by the fly ash.