## 3 ANN model  
### Development of ANN Model  
In this work, an ANN model for the isothermal compression flow behaviors of Inconel 718 superalloy in respect of dependency of strain, strain rate and temperature, was developed by taking deformation temperature (\(T\)), strain rate (\(\dot{\varepsilon}\)) and strain (\(\varepsilon\)) as input variables, and true stress (\(\sigma\)) as the only output variable [22]. The schematic representation of the ANN architecture was shown  
Figure 4: The metallographs under the conditions of (**a**) 1203 K and 0.01 s\({}^{-1}\) and (**b**) 1353 K and 0.01 s\({}^{-1}\). Figure 3: The stress-strain curves of Inconel 718 superalloy under different temperatures with the strain rates of (**a**) 0.01 s\({}^{-1}\); (**b**) 0.1 s\({}^{-1}\); (**c**) 1 s\({}^{-1}\) and (**d**) 10 s\({}^{-1}\). in Figure 5. In this network, all the continuous stress-strain curves measured from the isothermal compression tests were discrete with a strain interval of 0.01, and the discrete points corresponding to the strain of 0.05-0.9 were adopted as the primary data of ANN model. The total 1204 discrete data points from the 18 stress-strain curves except the four curves corresponding to 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\({}^{-1}\) were defined as the training data of the ANN model. Then, a total 344 data points on the other four curves were considered as the test data for the ANN to evaluate of the generalization performance, namely the adaptability of ANN model to fresh forming conditions beyond training. Subsequently, the corresponding 344 experimental data points were picked out and considered as the reference points for the test work. The test work would be conducted between the reference data and the test data. It is common that the determination of the appropriate number of hidden layers and neurons in each hidden layer is one of the most critical tasks in obtaining an accurate ANN model. It was assumed that the network structure with one or two hidden layers was adopted to test respectively, and then the appropriate number needed to be finally determined through the appropriate tolerance evaluation between the predicted and experimental data. An indicator of mean square error (_MSE_) as Equation (3) [23] was introduced to evaluate the training performance and the generalization performance of the ANN, and accordingly the optimal hidden layer number and neuron number would be determined by comparing the performance of different networks derived from different network structure parameters. It is noted that a smaller _MSE_-value indicates better network performance. As for two different network structures, i.e., one or two hidden layers, the relative _MSE_ plots along with the number of neurons in each hidden layer were calculated respectively and shown in Figure 6. The comparison of the _MSE_ plots between the different network structures in Figure 6 shows that the structure of two hidden layers induces lower _MSE_-value level, and thus it possesses higher generalization performance. Focusing on the cases with two hidden layers, the _MSE_-values decline with neuron number increasing, while it appears an inverse trend when the neuron number is above ten, which indicates that increasing neuron number up to 10 elevates the network performance significantly. Above all, two hidden layers and 10 neurons in each hidden layer were determined as the final network architecture. \[MSE=\frac{1}{N}\sum_{i=1}^{N}(E_{i}-P_{i})^{2} \tag{3}\]  
where \(E_{i}\) and \(P_{i}\) are respectively an experimental stress value and the corresponding predicted stress value; \(N\) is the number of data samples. Figure 5: The schematic representation of the artificial neural network (ANN) architecture. It is noteworthy that the values of the input and output variables distribute in distinct ranges and even dimensions, which induces poor convergence speed and prediction accuracy of ANN model.