## 4 Results and Discussion
### Results Evaluation
#### 4.2.2 Prediction Model for Flexural Strength  
The flexural strength the ANN model that was developed in this study was adopted to predict the flexural strength of UHPFRC. Comparisons between the predicted and experimental values for the training and testing data of the ANN model were shown in Figure 4. It was evident that the predicted values from the training and the testing data, as calculated by the ANN model, were closely to the target values. This phenomenon demonstrated that the ANN model could successfully learn the nonlinear relationship between the input and output variables. Therefore, the ANN model possessed potential to estimate the effects of steel fibers on the flexural strength of UHPFRC.  
The input layer weights (ILW), input layer bias (ILB), hidden layer weights (HLW), and hidden layer bias (HLB) of the flexural strength ANN model are given by Equations (18)-(21), respectively.  
Figure 4: Comparison of predicted values from flexural strength ANN model with experimental results: (**a**) Training data, (**b**) Testing data. The figures showed that the comparison between the target values and predicted values of testing and training data from experiments and flexural strength ANN model, the horizontal axis denotes number of training or testing data and the vertical axis denotes the flexural strength. It was obvious that the predicted values accord with the target values.  
\begin{table}
\begin{tabular}{c c c} \hline
**Indicators** & **Training** & **Testing** \\ \hline \(RMS\) & 0.0876 & 0.0980 \\ \(R^{2}\) & 0.9923 & 0.9901 \\ \(IAE\) & 0.0005 & 0.0019 \\ \hline \end{tabular}
\end{table}
Table 5: The indicators of training and testing of Compressive strength ANN model.  
\[\text{ILW}=\left[\begin{array}{cccc}-1.6618&-0.1478&-0.2359&0.3162&1.6211\\ -0.9701&0.8838&-0.8155&-1.0247&0.8334\\ 2.0128&-1.2745&-0.6188&-0.7977&-0.5454\\ 1.5968&0.7523&-0.0592&0.0347&-1.6347\\ 0.1462&-1.3740&1.5325&-0.5921&-2.0480\\ -1.7972&0.6656&0.6023&1.6868&-0.6638\\ -0.3699&-0.5348&0.3489&-1.7462&-0.1823\\ -1.0360&-1.0720&-1.1409&-1.4832&-0.7843\\ 0.5253&0.6616&2.7165&-1.4818&0.0997\\ 1.5109&1.9836&0.8304&-1.1222&0.6302\\ 1.7022&-2.0969&-1.8292&-1.1385&-0.8313\\ -1.2578&0.9580&0.3124&-0.0636&-1.2783\\ 0.8136&0.4598&-0.4208&2.2028&-1.5568\\ 1.9101&-0.0758&0.5641&-0.3651&-2.0535\\ 1.4933&0.4085&-0.9748&-0.5404&-1.6774\\ \end{array}\right] \tag{18}\]  
\[\text{ILB}=\left[\begin{array}{cccc}2.4280\\ 2.6112\\ 1.2367\\ -1.0888\\ -0.6916\\ 0.6397\\ 0.4543\\ 0.3855\\ 0.5287\\ 0.1019\\ -0.7088\\ -0.2661\\ 1.0694\\ 1.7951\\ 3.4860\\ \end{array}\right] \tag{19}\]  
\[\text{HLW}=\left[\begin{array}{cccc}0.0607&0.0399&-1.1485&-0.4111&-0.6869& \ldots\\ -0.2635&0.6018&-0.1671&0.4443&-1.5675&\ldots\\ -0.8053&0.5883&-0.4601&0.1114&-1.3230&\end{array}\right] \tag{20}\]  
\[\text{HLB}=\left[\begin{array}{cccc}0.0607&0.0399&-1.1485&-0.4111&-0.6869& \ldots\\ -0.2635&0.6018&-0.1671&0.4443&-1.5675&\ldots\\ -0.8053&0.5883&-0.4601&0.1114&-1.3230&\end{array}\right] \tag{21}\]  
\[\text{HLB}=\left[\begin{array}{cccc}0.0607&0.0399&-1.1485&-0.4111&-0.6869& \ldots\\ -0.2635&0.6018&-0.1671&0.4443&-1.5675&\ldots\\ -0.8053&0.5883&-0.4601&0.1114&-1.3230&\end{array}\right] \tag{22}\]  
\[\text{HLB}=\left[\begin{array}{cccc}0.0342\\ \end{array}\right] \tag{23}\]  
The performance indicators of the flexural strength ANN model for training and testing data, including \(RMS\), \(R^{2}\), and \(IAE\), were listed in Table 6. In Table 6, the indicators show that the proposed model produced good results. Thus, the flexural strength of the ANN model studied in this paper had high-precision and was suitable for evaluating the effects of steel fibers on the flexural strength of UHPFRC.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Indicators** & **Training** & **Testing** \\ \hline \(RMS\) & \(0.1492\) & \(0.0376\) \\ \(R^{2}\) & \(0.9777\) & \(0.9986\) \\ \(IAE\) & \(0.0011\) & \(0.0004\) \\ \hline \hline \end{tabular}
\end{table}
Table 6: The indicators of Training and Testing of Flexural strength ANN model.