## 3 Results and Discussions
### Analysis of Prediction Results
#### 3.2.2 Flow Behavior Analysis  
The number of hidden layers of the BPNN is one, and the number of neurons in the hidden layer is twelve. The population size is 30 and the maximum number of iterations is 200. To study the ANN for modeling the constitutive relations of Inconel 625, 220 experimental points in our experimentally obtained stress-strain curves were selected as the original data set (the training set selects 200 points and the test set selects 20 points) for the HT-1200 samples. The data points were selected based on the criterion that they should cover the complete stress-strain curve, including elastic, yield, reinforcement, and neck segments, so that the entire process of material tensile deformation can be completely simulated. In particular, the elastic limit, yield stress, and ultimate stress, which are key points that reflect the mechanical properties of the material, need to be included in the data set.  
Figure 4: Crystallographic orientation profiles of (**a**) AB sample and (**d**) HT-1200 sample; EBSD inverse pole figures (IPFs) of (**b**) AB sample and (**e**) HT-1200 sample; misorientation distribution of (**c**) AB sample and (**f**) HT-1200 sample; (**g**) proportion of deformed grains, substructured grains, and recrystallized grains for different samples.  
Figure 5: (**a**) True stress-true strain curves. The arrows in the inset indicate dimples. (**b**) comparison of the associated yield strength (YS), ultimate tensile strength (UTS), and elongation (EL).  
Figure 6 shows the comparison between the prediction results of different hybrid models and the actual data. For HT-1200, the prediction results of the EDBO-BPNN model are closer to the actual values in the initial region and the peak region of the stress-strain curve, indicating that the EDBP-BPNN model is superior to other comparison models in terms of prediction accuracy and can predict the actual experimental data more accurately. In the data of the later stage of the test, the stress of the fracture part has a sudden change, so the stress prediction of this part is the most difficult, yet the model proposed in this paper can still predict almost the same results as the test. In summary, among all the compared algorithms, the model proposed in this paper shows the best prediction performance in both the initial region of stretching and the region where the stress changes abruptly.  
Figure 7 visualizes the prediction evaluation metrics of all the hybrid models in the form of box-line plots to highlight the differences in the comparison of prediction performance. Among them, the EDBO-BPNN model outperforms the other models compared in all prediction evaluation metrics. The R\({}^{2}\) is 0.999964, RMSE is 1.1180, MAE is 0.5534, and MAPE is 0.1502% for the evaluated index of HT-1200. Table 1 presents the detailed data. With the same number of iterations, the EDBO-BPNN model can obtain better model parameters and thus achieve higher prediction accuracy.  
In general, the predicted results of the EDBO-BPNN agree best with the experimental results and can well predict the tensile stress of the Inconel 625 superalloy manufactured by WAAM. It should be noted that although all the hybrid BPNN models improve the prediction accuracy of the BPNN model by optimizing the initial parameters of the BPNN, the EDBO-BPNN proposed in this study achieves a better balance between local and global search, and thus has a significant advantage in optimizing the BPNN model.  
Figure 6: Comparison of the experimental versus different hybrid models’ predicted values.