## 5 Application  
During the hot deformation process, the material would experience extensive temperature and strain rate regions. Thus, the accurate numerical simulation of hot forming process naturally needs a great deal of stress-train data in wide deformation conditions. However, the measurement of experimental data is undoubtedly a time-consuming and effort-consuming task. In such a situation, the accurate prediction of stress-strain data outside of experimental conditions can actually help a lot. In this investigation, the ANN model was implanted in a finite element solver with the interface of 'URPFLO' subroutine of Fortran language. The prediction arithmetic of the ANN model was presented in Equations (8) and (11) [25], according to which the program codes implanting the ANN model into FE solver were written and displayed in the final Appendix A part. The input \(net_{j}\) of \(j\)th node in hidden layer:  
\[net_{j}=\sum_{i=1}^{l}w_{ij}x_{i}+a_{j} \tag{8}\]  
The output \(y_{j}\) of \(j\)th node in hidden layer:  
\[y_{j}=\phi(net_{j})=\phi(\sum_{i=1}^{l}w_{ij}x_{i}+a_{j}) \tag{9}\]  
The input \(net_{k}\) of \(k\)th node in output layer:  
\[net_{k}=\sum_{j=1}^{l}w_{ijk}y_{j}+b_{k}=\sum_{i=1}^{q}w_{jk}\phi\left(\sum_{i =1}^{l}w_{ij}x_{i}+a_{j}\right)+b_{k} \tag{10}\]  
The output \(z_{k}\) of \(k\)th node in output layer:  
\[z_{k}=\psi(net_{k})=\psi\left(\sum_{j=1}^{l}w_{jk}y_{j}+b_{k}\right)=\psi\left( \sum_{i=1}^{q}w_{ki}\phi\left(\sum_{i=1}^{l}w_{ij}x_{i}+a_{j}\right)+b_{k}\right) \tag{11}\]  
Figure 10: The continuous mapping relationship among temperature, strain rate, strain and stress (**a**) in a global version as well as under fixed (**b**) temperatures; (**c**) strain rates and (**d**) strains. where \(x_{i}\) is the \(i\)th input vector; \(i,j\) and \(q\) are respectively the input, hidden and output layer neuron numbers; \(a_{j}\) and \(b_{k}\) are respectively the \(j\)th and \(k\)th neuron bias of hidden layer and output layer; \(w_{ij}\) and \(w_{jk}\) are respectively the weights transferring from the \(i\)th neuron of output layer to the \(j\)th neuron of hidden layer, and the \(j\)th neuron of hidden layer to the \(k\)th neuron of output layer; \(net_{k}\) and \(net_{j}\) are respectively the received values of the \(k\)th neuron for output layer and the \(j\)th neuron for hidden layer; \(\psi\) and \(\phi\) are respectively the transfer functions for the output layer and hidden layers; \(z_{k}\) and \(y_{j}\) are respectively the outcomes of output layer and hidden layers. The FE solver inserted ANN model was employed to simulate the isothermal compression tests of Inconel 718 superalloy under the conditions of 1153 K and 0.1 s\({}^{-1}\), 1303 K and 0.1 s\({}^{-1}\), 1153 K and 1 s\({}^{-1}\) and 1253 K and 1 s\({}^{-1}\). The established FE model was presented in Figure 11. In the FE model, the anvils and specimen were respectively defined as rigid objects and plastic object. During the actual isothermal compression process in Gleeble 3500 test machine, constant temperature is qualitatively ensured for the specimen, so the heat transfer and thermal radiation of the specimen with the anvils and environment as well as the work-heat conversion were ignored in the present FE model. Besides, a constant shear friction coefficient of 0.1 was set to conform to the practical graphite lubricant on the contact surfaces between the anvils and specimen. To meet the demand of fixed strain rates, the compression velocity of the driver anvil (anvil I) was defined according to Equation (12). The compression rate was 60% corresponding to 7.2 mm height reduction of the specimen. \[v=h_{0}\hat{e}\exp(-\hat{e}t) \tag{12}\]  
where \(v\) stands for the instantaneous compression velocity of the driver anvil; \(h_{0}\) is the original height of the specimen, and here \(h_{0}=12\) mm; \(\hat{e}\) is strain rate and \(t\) is the time point. The effective strain and effective strain rate distributions of the specimens sectioned along the axial direction were presented respectively in Figures 12 and 13.