# Methodology for the Prediction of the Thermal Conductivity of Concrete by Using Neural Networks
## 4 Results and Discussion
### Integration of Both Models to Predict Thermal Conductivity  
The third phase of this study is dedicated to examining the impact of synthetic data on the training of a neural network model. This evaluation is conducted to ascertain whether a model trained with augmented data can produce results comparable to the performance achieved through training with real data and possibly enhance it. In order to assess this impact, this study investigates three distinct scenarios: (a) training the MLP model exclusively with the real dataset containing 200 entries, (b) training the model with \begin{table}
\begin{tabular}{c c} \hline \hline
**Column** & **Score** \\ \hline Density & 0.9133 \\ w/c ratio & 0.7806 \\ Water & 0.8061 \\ Cement & 0.8827 \\ Ceramic powder & 0.5408 \\ Fine agg. & 0.8673 \\ Coarse agg. & 0.6684 \\ Natural agg. & 0.5153 \\ Recycled agg. & 0.8622 \\ Light agg. & 0.3878 \\ Waste agg. & 0.6480 \\ Fly ash & 0.4796 \\ Silica fume & 0.6480 \\ Slag & 0.5357 \\ Fiber & 0.5918 \\ Admix. & 0.5255 \\ Superplast. & 0.9235 \\ Foam volume & 0.5000 \\ tc & 0.8827 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: KS Test evaluation. the synthetic data consisting of 200 entries first and subsequently incorporating the real dataset, and (c) training the model with 1000 data points of the synthetic data and then integrating the real dataset. The same hyperparameters utilized in developing the final model are maintained in this integration process. The key distinction lies in the dataset used for the training and validation phases. Our approach initially involves training the model first with synthetic data. Subsequently, the same model architecture uses the real dataset to enhance the model's performance. The results for the previous MLP model (a) and scenarios (b) and (c) are summarized in Table 6, indicating the outcomes of the final model. The predictive capabilities of the developed model are thoroughly examined using a new dataset comprising 30 data points to assess its generalization performance. Despite the limited size of the dataset in scenario (a), the training and validation phases yield commendable results, with an RMSE of 0.0604 and an R\({}^{2}\) of 0.9846. In scenarios (b) and (c), where synthetic data play a vital role, both demonstrate exemplary performance during training, achieving R\({}^{2}\) values of 0.9995 and 0.9962. This improvement in the results also extends to the validation dataset, with R\({}^{2}\) values of 0.9938 and 0.9754. Post-incorporation of real data into the training process, the metrics exhibit comparable values during training and improvements in validation, 0.9870 and 0.9960 for scenarios (b) and (c), respectively. Upon rigorous evaluation of the independent test set, scenario (c) emerges as the most robust and indicates the most favorable combination of performance metrics. The predictive performance of the developed MLP model for the three scenarios is evaluated on an independent test dataset that can be graphically seen in Figures 8 and 9. Comparing the actual and predicted thermal conductivity results for each scenario, a distinct pattern emerges, underscoring the model's significant performance enhancement when integrated with synthetic data. In Figure 8a, we present the predictions for the first case, wherein a considerable portion of the points align with the ideal x = y line. However, some deviations are noticeable, with errors exceeding 0.12. Figure 8b,c illustrate scenarios where the model is trained using synthetic data. In both instances, incorporating synthetic data leads to a discernible improvement in prediction accuracy and a notable error reduction. Scenario (c) stands out, boasting a larger dataset and delivering the most compelling outcomes. This scenario achieves the optimal combination of performance metrics, represented by a best-fit line with an impressive R\({}^{2}\) value of 0.9975 and a minimal RMSE of 0.0244.