## 3 Results and Discussion
### Machine Learning to Predict \(q_{u}\) and \(E_{t}\)
#### 3.4.1 Comparison of Machine Learning Models
In the test set, both the ANN and ANN-GMDH models exhibit consistent performance, with the ANN model achieving an R-squared of 0.9944, an MAE of 18.9, and an RMSE of 27.63, while the ANN-GMDH model achieves an R-squared of 0.9949, an MAE of 18.45, and an RMSE of 26.37. This performance  
reiterates the models' proficiency in generating dependable predictions on independent data instances. When juxtaposed with SR model, it becomes apparent that ANN-GMDH and ANN models occupy a favorable position. The models aptly capture underlying relationships within the data and translates them into reliable predictions. This performance is poised to deliver valuable insights for compressive strength prediction applications. While both the ANN-GMDH and ANN models achieve almost the same metrics, the ANN-GMDH model achieves better accuracy overall. However, it is important to note that its \(r_{d}\) value is slightly worse than that of the ANN model. Figure 16: The cross plots for actual and predicted \(\frac{Et}{Ia}\) targets of training, validation, test, and combined data for ANN (\(E_{t}\)) model and ANN-GMDH (\(E_{t}\)) model: (**A**) ANN (\(E_{t}\)) model for training data, (**B**) ANN (\(E_{t}\)) model for validation data, (**C**) ANN (\(E_{t}\)) model for test data, (**D**) ANN (\(E_{t}\)) model for combined data, (**E**) ANN-GMDH (\(E_{t}\)) model for training data, (**F**) ANN-GMDH (\(E_{t}\)) model for validation data, (**G**) ANN-GMDH (\(E_{t}\)) model for test data, and (**H**) ANN-GMDH (\(E_{t}\)) model for combined data. Figure 17: Comparison of true and predicted \(q_{u}\) targets in training, validation, test, and combined data for ANN (\(q_{u}\)) model and ANN-GMDH (\(q_{u}\)) model: (**A**) ANN (\(q_{u}\)) model for training data, (**B**) ANN (\(q_{u}\)) model for validation data, (**C**) ANN (\(q_{u}\)) model for test data, (**D**) ANN (\(q_{u}\)) model for combined data, (**E**) ANN-GMDH (\(q_{u}\)) model for training data, (**F**) ANN-GMDH (\(q_{u}\)) model for validation data, (**G**) ANN-GMDH (\(q_{u}\)) model for test data, and (**H**) ANN-GMDH (\(q_{u}\)) model for combined data. Moreover, establishing an acceptable error margin is pivotal in assessing the sufficiency of the training dataset. Figures 16 and 17 depict a 20% error margin on each cross plot to illustrate this concept. The performance of both the \(E_{t}\) and \(q_{u}\) models was commendable, as the majority of predictions fell within the error margin lines across the training, validation, and testing phases. This underscores the adequacy of the training dataset in supporting the predictive models for \(E_{t}\) and \(q_{u}\) using the proposed ANN model in this study. The precision observed in predicted accuracy is the result of a strategy based on statistical rigor and practical relevance. The models were trained to detect subtle patterns in the data by carefully selecting variables and exhaustively assessing their statistical features. Additionally, a diverse array of machine learning methods including ANN, GBR, XGBoost, SVR, DTR, and RFR were employed to ensure comprehensive exploration of predictive patterns. Furthermore, the intensive evaluation approach, which included cross-validation techniques on 225 datasets, enabled a thorough assessment of the models' generalization capabilities. Hyperparameter tuning fine-tuned the models' performance, ensuring they were well-suited to the dataset's subtleties. Recognizing the inherent randomness and possibility for human error, the methodology promoted openness and accountability by employing statistical indicators such as RMSE, MAE, and R2. This comprehensive approach emphasizes the methodology's rigor and the predictive models' reliability.