## 3 Model Construction and Prediction Results
### Comparison and Evaluation of Empirical Formulas  
To verify the accuracy and reliability of the model presented in this paper, three classic empirical formulas for concrete failure criteria commonly used in practice were compared and analyzed, namely, the following:  
(1) The formula for calculating the compressive strength of SFRC in China's _Code for Design of Concrete Structures_ (GB 50010-10) [23] is as follows:  
\[f_{cu}=kf_{c0}+aV_{f} \tag{6}\]  
(2) The American Concrete Institute's _Guide to Design with Fiber-Reinforced Concrete_ (ACI 544.4R-18) [24] provides specifications and design considerations for SFRC:  
\[f_{cu}=0.97f_{c0}V_{m}+494V_{f}l/d \tag{7}\]  
Figure 3: The performance of four kinds of models on the training and testing sets: (**a**) DNN model; (**b**) RF model; (**c**) XGB model; (**d**) BP model.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{4}{c}{**Training Set**} & \multicolumn{4}{c}{**Test Set**} \\ \cline{2-10}  & \(\mathbf{R^{2}}\) & **MAPE** & **RMSE** & **MAE** & \(\mathbf{R^{2}}\) & **MAPE** & **RMSE** & **MAE** \\ \hline DNN & 0.9943 & 2.58\% & 2.95 & 2.22 & 0.9398 & 8.16\% & 9.41 & 6.62 \\ RF & 0.9892 & 3.04\% & 3.93 & 2.63 & 0.9179 & 10.33\% & 11.17 & 7.58 \\ XGBoost & 0.9417 & 8.27\% & 9.11 & 6.81 & 0.8804 & 11.31\% & 13.39 & 9.77 \\ BP & 0.9593 & 6.40\% & 7.79 & 4.35 & 0.8900 & 9.63\% & 12.32 & 8.86 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Model performance evaluation.  
(3) CEB-FIP [25] recommended the following calculation formula:  
\[f_{cu}=kf_{c0}+aV_{f}+bV_{f}^{2} \tag{8}\]  
where \(f_{\text{cu}}\) is the compressive strength of steel fiber reinforced concrete;\(f_{c0}\) is the compressive strength of ordinary concrete, \(f_{c0}=\frac{A}{B^{u/c}}\big{(}1+\frac{t}{28}\big{)}\); \(V_{f}\) is the steel fiber volume fraction; \(V_{\text{m}}=1-V_{f}\); and \(k\), \(a\), \(b\) are material parameters, required experimental data fitting.  
These three classical and widely used SFRC strength prediction models are used to carry out numerical calculations on the testing set data one by one. The numerical calculation results of each strength prediction model and the prediction results of the deep learning model in this study are consistent with the real test data, as shown in Figure 4. According to the visual analysis in Figure 4, the model proposed by the Chinese and American specifications is consistent, but with large errors, while the C-F model demonstrates the lowest accuracy. However, the dispersion of compressive strength predicted by the DNN model is smaller than that of these three classical strength prediction models, indicating better accuracy and reliability. It can be concluded that the DNN model established in this study can predict the compressive strength of SFRC more efficiently than the classical strength prediction model, and has better applicability. In addition, Table 4 shows the evaluation indicators on the testing set of the DNN model with these three classical strength formulas. According to the performance of each evaluation index, the DNN model exhibits superior accuracy and stronger predictive capability compared to classical strength formulas.  
Figure 4: The performance of four kinds of models on the training and testing sets: (**a**) DNN model; (**b**) GB 50010 model; (**c**) ACI 544.4R model; (**d**) CEB-FIP model.