# Methodology for the Prediction of the Thermal Conductivity of Concrete by Using Neural Networks
## 4 Results and Discussion
### Development of the Multilayer Perceptron Model  
The primary objective of this work is to develop an MLP model for predicting the thermal conductivity of concrete based on its mass composition and density. The hyperparameters of the MLP model are initially predefined and subsequently fine-tuned to enhance the prediction accuracy. Table 3 provides an overview of the model's key features, encompassing the hyperparameters, evaluation metrics, activation functions, optimization methods, and loss functions. To establish the optimal architecture of the MLP model, the number of hidden layers and neurons is systematically varied, and a combination of the metrics RMSE and R\({}^{2}\) is employed to measure the model's overall performance. A diligent trial-and-error approach is undertaken to determine the most effective configuration for the number of hidden layers and neurons in each layer. Table 4 displays the loss functions and performance metrics of the multiple neural networks assessed in this study. Notably, the most proficient model features a two-hidden-layer structure with 200 and 100 neurons, yielding an RMSE of 0.111 W/m.K and an R\({}^{2}\) of 0.984 for the training dataset. As for the validation dataset, the model demonstrates metrics values of 0.183 and 0.960, respectively. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Layers** & **ANN Architecture** & **Train Loss** & **Valid Loss** & **RMSE** & **R\({}^{2}\)** \\ \hline
4 & 200–100 & 0.083 & 0.012 & 0.111 & 0.984 \\
5 & 200–100–50 & 0.107 & 0.024 & 0.155 & 0.968 \\
5 & 200–100–40 & 0.099 & 0.037 & 0.192 & 0.950 \\
5 & 200–100–30 & 0.087 & 0.028 & 0.167 & 0.960 \\
5 & 200–100–20 & 0.012 & 0.037 & 0.191 & 0.949 \\
6 & 200–100–50–25 & 0.102 & 0.064 & 0.253 & 0.918 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Performance indices and metrics of the MLP model. \begin{table}
\begin{tabular}{l l} \hline \hline
1. Hyperparameters & Hidden layers = [1, 2, 3, 4] \\  & Batch size = 64 \\  & Learning rate = 0.001 \\  & Epochs = 100 \\ \hline
2. Metrics & RMSE \\  & R\({}^{2}\) \\ \hline
3. Activation function & ReLU \\ \hline
4. Optimization function & Stochastic Gradient Descent (SGD) \\ \hline
5. Loss function & MSE \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Main features of the MLP model. Regarding the best-performing model, an analysis of the loss functions and the performance metrics versus the number of epochs is conducted (Figure 6). As the number of epochs progresses, a significant pattern appears in the training and validation processes. Both training and validation losses exhibit an exponential decrease as iterations continue. In parallel, the performance metrics on the validation dataset align with the behavior of the loss function. While RMSE steadily declines, \(R^{2}\) shows a corresponding increase, achieving its minimum and maximum value at approximately 100 epochs. This observed trend reflects the model's capacity to optimize its predictive accuracy as it refines its learning over successive epochs. Comparing the findings of the MLP model to previous studies, they demonstrate a remarkable level of accuracy for the training and validation. A study by [PERSON] et al. [28], using a dataset with 132 entries, developed an ANN based on the mechanical properties and the density of concrete, which had an overall \(R^{2}\) performance of 0.996. However, the ANN architecture used only one hidden layer with 5 to 25 neurons. [PERSON] et al. [6] took a step further by implementing a neural network with a larger dataset, with an RMSE of 0.117 and \(R^{2}\) of 0.964 for the training set and an RMSE of 0.215 and \(R^{2}\) of 0.894 for the validation set. Nonetheless, large residuals were obtained. Another high-performance ANN model was developed by [PERSON] et al. [31] where they investigated the influence of concrete composition and temperature on thermal conductivity. Despite the results of an \(R^{2}\) of 0.998 and RMSE of 0.005 during neural network training, only 12 inputs were used.