## 3 Results and Discussion  
### Machine Learning-Based Prediction of Compressive Strength  
Based on the performance data of machine learning models in predicting compressive strength (Fc), a comprehensive evaluation of various algorithms was conducted. Figure 5 illustrates the relationship between experimentally measured values and machine learning (ML) predictions. In the training phase, NNR exhibited the best overall performance, achieving an R\({}^{2}\) value of 0.9869, the highest among all models. Both RF and GPR also showed excellent predictive capabilities, with training R\({}^{2}\) values of 0.9852 and 0.9802, respectively. In contrast, KNNR delivered the lowest correlation coefficient on the training set ([PERSON] et al., 2016). During the testing phase, the NNR model retained its superior performance, with a test R\({}^{2}\) of 0.9855 and minimal decline compared to the training set, indicating strong generalization ability ([PERSON] et al., 2016). As shown in Figure 5, the NNR model predictions align closely with the diagonal reference line, indicating high accuracy and minimal bias. In contrast, the KNNR and LMR models display noticeable dispersion from the ideal line, particularly at the lower and upper extremes of compressive strength values, suggesting reduced generalization capacity. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Evaluation** & **RMSE (Root Mean** & **MAE (Mean Absolute** & **MAPE (Mean Absolute** & **R\({}^{2}\) (Coefficient of** \\
**Parameters** & **Squared Error)** & **Error)** & **Percentage Error)** \\ \hline Equation & \(RMSE=\sqrt{\frac{\sum\limits_{i=1}^{n}{(y_{i}-y_{i})^{2}}}{n}}\) & \(MAE=\frac{\sum\limits_{i=1}^{n}{(y_{i}-y_{i})^{2}}}{n}\) & \(MAPE=\frac{100\%}{n}\sum\limits_{i=1}^{n}{\left|\frac{y_{i}-y_{i}}{y_{i}} \right|}\) & \(R^{2}=1-\frac{\sum\limits_{i=1}^{n}{(y_{i}  
The variation in performance between the training and testing models reveals key insights into their generalization characteristics [97]. Although RF reached a high training R\({}^{2}\) of 0.9852, its test R\({}^{2}\) dropped to 0.9625, suggesting a tendency toward overfitting. GPR showed slightly better performance on the test set (R\({}^{2}\) = 0.9733) compared to the training set (R\({}^{2}\) = 0.9802), which may imply suboptimal model convergence during training. Overall, NNR and DTR demonstrated the most stable and reliable predictive performance [98], particularly in the test phase, making them favorable choices for engineering applications involving compressive strength prediction. Future research should focus on parameter optimization and dataset expansion to validate their robustness. Statistically, the NNR model consistently delivered top-tier performance on both training and testing sets. It achieved exceptional R\({}^{2}\) values of 0.9869 and 0.9855 under five-fold cross-validation, confirming its advantage [99,100]. Figure 6 offers a detailed comparison of the predicted and experimental compressive strengths. While the models exhibited high accuracy within the primary strength range of 20-50 MPa, significant systematic bias was observed in extreme regions. In the high-strength domain (>50 MPa), predictions were overestimated by approximately 5 MPa. Meanwhile, in the low-strength domain (<20 MPa), predictions were underestimated by approximately 2 MPa on average. These discrepancies may arise from several factors: (a) insufficient data samples in the extreme strength regions, which accounted for only 12% of the total dataset; (b) limited sensitivity of the models to the nonlinear hardening behavior of foamed concrete; and (c) excessive noise introduced by less influential input features [101,102]. Figure 6 highlights the absolute prediction errors between the measured and ML-predicted values. Figure 7 shows that NNR and DTR produce the lowest absolute errors across most data points, while RF and KNN show larger error spikes, which may be due to their sensitivity to local variations or overfitting to small clusters of data. Figure 5: Regression graph for compressive strength.