# Viscosity Prediction of Different Ethylene Glycol/Water Based Nanofluids Using a RBF Neural Network
## 4 Results and Discussion  
For the RBF neural network, the Spread is usually a very important factor influencing the training process. Figure 4 shows the relationships of the mean square error (MSE) and the number of hidden layer neurons with different Spreads. Analyzing the results reported in Figure 4, it is found that for the same converged target, the neuron numbers in the hidden layer need to be increased obviously with the decrease of the Spread. When the Spread varies from 1 to 0.1, the corresponding neuron configuration of the RBF neural network are 5-38-1, 5-40-1, 5-56-1, 5-67-1, and 5-105-1, respectively. With the decrease of the Spread, the CPU time for computing the RBF neural network will also increase. At a Spread of 1, 0.5, 0.3, 0.2, and 0.1, the corresponding CPU times are 6.318, 6.396, 8.798, 10.827, and 15.772 s, respectively. In addition, Table 2 lists the values of four evaluation criteria for predicting the viscosity ratio of ethylene glycol/water based nanofluids by using the RBF neural network with different Spreads. It can be seen from Table 2 that although all \(R^{2}\) are within the acceptable level of 0.99, the prediction performance of the RBF neural network is still affected by the value of the Spread, especially for the testing samples. Based on the comprehensive considerations of modeling complexity, prediction accuracy, and CPU time, the RBF neural network with the neuron configuration of 5-56-1 and Spread of 0.3 is used in this study. The related weights and biases of the 5-56-1 RBF neural network can be found in Table 3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Object**} & \multirow{2}{*}{
\begin{tabular}{c} **Evaluation** \\ **Criteria** \\ \end{tabular} } & \multicolumn{5}{c}{**Spread**} \\ \cline{3-7}  & & **1** & **0.5** & **0.3** & **0.2** & **0.1** \\ \hline \multirow{4}{*}{Training samples} & _RMSE_ & 0.04651 & 0.04460 & 0.04630 & 0.04502 & 0.04400 \\  & _MAPE_ (\%) & 2.3335 & 2.27321 & 2.09967 & 1.93784 & 2.00530 \\  & _SSE_ & 0.42829 & 0.39383 & 0.42454 & 0.40125 & 0.38337 \\  & \(R^{2}\) & 0.99925 & 0.99932 & 0.99927 & 0.99930 & 0.99934 \\ \hline \multirow{4}{*}{Testing samples} & _RMSE_ & 0.09190 & 0.07035 & 0.04443 & 0.05124 & 0.10061 \\  & _MAPE_ (\%) & 4.65795 & 3.90042 & 2.43228 & 2.67720 & 4.27471 \\  & \(SSE\) & 0.15201 & 0.08908 & 0.03553 & 0.04727 & 0.18221 \\  & \(R^{2}\) & 0.99590 & 0.99760 & 0.99904 & 0.99872 & 0.99508 \\ \hline \multirow{4}{*}{Total samples} & _RMSE_ & 0.05183 & 0.04728 & 0.04615 & 0.04557 & 0.05117 \\  & _MAPE_ (\%) & 2.52721 & 2.40881 & 2.12738 & 1.99945 & 2.19442 \\ \cline{1-1}  & \(SSE\) & 0.58030 & 0.48291 & 0.46007 & 0.44852 & 0.56558 \\ \cline{1-1}  & \(R^{2}\) & 0.99906 & 0.99921 & 0.99925 & 0.99927 & 0.99908 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance evaluation of RBF neural networks with different Spreads. Figure 4: The relationships of mean square error (MSE) and the number of hidden layer neurons with different Spreads. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**Hidden Layer**} & \multicolumn{6}{c}{**Output Layer**} \\ \cline{2-9}
**Neuron** & \multicolumn{6}{c}{**Weights (\(w_{ij}\)) a and Biases**} & \multicolumn{6}{c}{**Weights (\(w_{ij}\)) b and Biases**} \\ \cline{2-9}  & \(T\) & \(d_{p}\) & \(\phi_{P}\) & \(\rho_{P}\) & \(H_{f}\) & **Biases** & \(\mu_{p}\)/\(H_{f}\) & **Biases** \\ \hline
1 & 0  
Figure 5 compares the predicted viscosity ratio of the RBF neural network and the experimental data involving the training and testing samples. It can be seen that all the prediction errors of the RBF neural network are within the \(\pm 10\%\) error bands.