# 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
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Scenario**} & \multirow{2}{*}{**Dataset**} & \multicolumn{2}{c}{**Overall**} \\ \cline{3-4}  & & **RMSE** & **R\({}^{2}\)** \\ \hline \multirow{2}{*}{(a)} & Training & 0.0443 & 0.9923 \\  & Validation & 0.3233 & 0.9228 \\  & Test & 0.0604 & 0.9846 \\ \hline \multirow{2}{*}{(b)} & Training & 0.0221 & 0.9980 \\  & Validation & 0.1329 & 0.9870 \\  & Test & 0.0439 & 0.9919 \\ \hline \multirow{2}{*}{(c)} & Training & 0.0219 & 0.9981 \\  & Validation & 0.0735 & 0.9960 \\  & Test & 0.0244 & 0.9975 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Performance metrics of the MLP model trained with both datasets. Figure 9 shows the relationship between the actual values derived from the test dataset from published papers and the corresponding predictions generated by the Multilayer Perceptron (MLP) model. This visual representation allows us to assess the model's accuracy in forecasting thermal conductivity, providing valuable insights into its performance. In Figure 9a, we depict the model utilizing solely the real dataset, while Figure 9b,c display the model trained with both the 200-point and 1000-point datasets, respectively. Although Figure 9a reveals a close alignment between predicted and real points, some discrepancies across various data points with errors as high as 0.13 and 0.14 W/m.K observed. Moving to Figure 9b, a significant reduction in these discrepancies becomes evident, especially within the low thermal conductivity values spanning from 0.5 to 1.0 W/m.K in range. This evidence signifies a marked improvement in prediction accuracy, resulting in a substantial decrease in both the number of points with errors and the magnitude of these errors, reducing the maximum error to 0.11 W/m.K. Figure 9c unveils an even more prominent enhancement in accuracy, with predictions exhibiting a higher level of precision and a further reduction in errors. In this case, the most significant error recorded is 0.05 W/m.K. Figure 8: Performance and predictions of the thermal conductivity of the test dataset for the three scenarios: (**a**) training the model with real data only, (**b**) training the model with 200 synthetic data points and real data, and (**c**) training the model with 1000 synthetic data points + real data. Figure 9: Comparison of actual thermal conductivity with model predictions across different training scenarios: (**a**) training the model with real data only, (**b**) training the model with 200 synthetic data points and real data, and (**c**) training the model with 1000 synthetic data points + real data. In summary, an impressive alignment between the actual and predicted values is achieved across all three cases. The first case exhibits a comparatively lower performance than the other two, potentially attributed to the limited size of the dataset. The presence of a smaller database may result in a more significant deviation between actual and predicted values, particularly within the thermal conductivity range of 0.5-1.0 W/m.K. This scarcity of data becomes apparent when synthetic data are introduced for training, ultimately enhancing the model's accuracy in predicting the thermal conductivity property. The methodology employed in this study endeavors to show the efficacy of neural networks in predicting the thermal properties of concrete based on the mass composition of its constituents. Furthermore, it seeks to investigate the utilization of synthetic data generation for training neural networks, addressing the challenge of data scarcity, a significant impediment to the accuracy of models employing machine or deep learning techniques. While the existing literature demonstrates notable success in utilizing neural networks for thermal conductivity prediction, our model achieves higher accuracy and incorporates a more extensive array of concrete constituents into the modeling process. This diversity enhances the model's versatility, expanding its applicability to a broader range of materials.