## 2 Methods
### Training and Prediction of the Hybrid Network  
The data preprocessing phase includes dataset creation and multi-dimensional normalization. In dataset creation, relevant logging parameters are first selected as inputs, and input sequences are generated using a sliding window. Normalizing the logging data is essential to ensure data consistency and processing efficiency. The multidimensional normalization method is employed to scale the data to a specific range and distribution, facilitating improved learning and processing by the model. Assume that the original data feature \(x\) takes the range of values of \(x-x_{min}\), normalized to the range \([a,b]\), where \(a\) and \(b\) are the minimum and maximum values of the target normalized range, \(x_{min}\) is the minimum value of the feature in the dataset, \(x_{max}\) is the maximum value of the feature in the dataset, and \(y\) is the result after multidimensional normalization. The normalization formula is described as:  
\[y=a+\frac{(x-x_{min})\times(b-a)}{(x_{max}-x_{min})} \tag{4}\]  
The proposed hybrid network is trained using Adam optimizer and the mean square error (MSE) loss function. Throughout training, the network iteratively optimizes its parameters to learn the relationship between input sequence features and target values, leveraging the specified optimizer and loss function. The validation set is used to evaluate model performance and prevent overfitting during training.  
Figure 5: The structure of the Inception–attention–BiLSTM hybrid network.  
After training, the trained network is deployed to predict outcomes on new data. The test data is prepared through pre-processing to generate input sequences, which are subsequently inputted into the proposed network for prediction using its designated prediction method. During prediction, the model computes predicted outputs based on learned weights and input data features. Post-processing techniques such as inverse normalization can be employed on the prediction results to generate final predicted values. This structured methodology reinforces the network's effectiveness and generalization capability throughout the training and prediction phases.  
Otherwise, Mean Absolute Error (MAE) is applied as a loss function to measure the differences between predicted values and true values. MAE is calculated as follows:  
\[MAE=\frac{1}{n}{\sum_{i=1}^{n}}|\hat{y}_{i}-y_{i}| \tag{5}\]  
where \(n\) is the number of samples, \(\hat{y}_{i}\) is the value of the \(i\)-th sample predicted by the network, and \(y_{i}\) is the true value of the \(i\)-th sample.  
The correlation coefficient (R\({}^{2}\)) is used to measure the fit of the regression model. R\({}^{2}\) is calculated as follows:  
\[R^{2}=1-\frac{\text{SS\_res}}{\text{SS\_tot}} \tag{6}\]  
\[\text{SS\_res}={\sum_{i=1}^{n}}{\Big{(}y_{i}-\hat{y}_{i})^{2}} \tag{7}\]  
\[\text{SS\_tot}={\sum_{i=1}^{n}}{\Big{(}y_{i}-\overline{y}_{i})^{2}} \tag{8}\]  
where SS_res (Sum of Squares Residuals) represents the sum of the squares of the errors between the model's predicted values and the true value, SS_tot (Total Sum of Squares) represents the total variation between the dependent variable and its mean, \(y_{i}\) is the true value, \(\hat{y}_{i}\) is the predicted value, \(n\) is the sample size, and \(\overline{y}_{i}\) is the mean of predicted values.  
The workflow of the Vs prediction is shown in Figure 6.  
Figure 6: The Vs prediction workflow of the proposed hybrid network.