# Methodology for the Prediction of the Thermal Conductivity of Concrete by Using Neural Networks
## 4 Results and Discussion
### Development of the Generative Adversarial Network Model  
The second part of the study considers the implementation of CopulaGAN for data augmentation. As can be observed in Table 4, increasing the number of layers does not improve the model's learning capacity or performance. If we increase the hidden layers of the model with a small dataset, the model can overfit. For this reason, the loss functions exhibit higher values. In order to solve this issue, we intend to develop a GAN model to increase the dataset, creating new synthetic data based on the real data used to develop the MLP model. This will avoid the overfitting process and make the model more robust with extensive dataset training. After generating the synthetic data, two aspects of the original data were evaluated: the column shapes and pair trends. Each aspect contributed differently to the overall score, which reached 78.14%, indicating that the synthetic data has acceptable quality. In Figure 7, we present the distribution of the augmented dataset. It illustrates the distribution of normalized thermal conductivity values for both real and synthetic datasets. The real data, shown in dark blue, have a sharper peak, while the synthetic data, depicted in light blue, exhibit a similar peak but with a slightly different spread. The overlap between the two distributions indicates that the synthetic data generated by CopulaGAN are largely representative of the real data, despite some discrepancies. After the synthetic data generation, we aim to incorporate it into the MLP model for joint training with real and artificial datasets. To assess the model's response to this diverse training approach, two distinct sets of synthetic data were created, one comprising 200 data points and the other consisting of 1000 data points. The seamless integration of these two neural network models and the subsequent validation of the model using a new test set are elaborated upon in the forthcoming section. Statistical Evaluation of the Generative Adversarial Network Model  
To assess the quality of the synthetic data, two metrics were utilized: the Correlation Matrix Comparison and the Kolmogorov-Smirnov Test (KS Test). The Correlation Matrix Comparison examines whether the relationships between variables in the synthetic dataset are consistent with those in the real dataset. The KS Test, on the other hand, compares the distributions of the two datasets to determine if they are drawn from the same underlying distribution. According to Table 5, several key observations can be made. Variables with high KS scores, such as density (0.913), cement (0.883), superplasticizer (0.923), and thermal conductivity (tc) (0.883), indicate that the synthetic data closely resemble the real data for  
Figure 7: Distribution of the synthetic generated by GAN and the real data. these variables. Moderate scores, including water/cement ratio (0.781), water (0.806), and fine aggregate (0.867), show good correspondence, though the match is slightly less precise. However, lower scores for variables like ceramic powder (0.541), natural aggregate (0.515), fly ash (0.480), and light aggregate (0.388) suggest that the synthetic data do not replicate these distributions as effectively, highlighting potential areas for improvement in the data generation model. In the Correlation Matrix Comparison, the analysis focuses on the similarity between the correlations in the real and synthetic datasets. A score closer to 1 indicates a stronger resemblance. The majority of variable pairs have high correlation similarity scores, typically above 0.95, signifying that the synthetic data effectively mirror the correlation structure of the real data. For example, the correlation between density and cement shows a score of 0.979, with real and synthetic correlations being \(-0.118\) and \(-0.078\), respectively, suggesting that the synthetic data closely capture this relationship. However, while the correlation similarity scores are generally high, there are differences in the exact correlation values between some pairs.