# A Multiscale CNN-Based Intrinsic Permeability Prediction in Deformable Porous Media
## 5 Model (2): Informed \(K^{5}\)-CNN Model  
Following the study of [PERSON] et al. [26] to improve the accuracy of the CNN model, in this section, we test an alternative structure to our \([K^{5}-n^{F}]\)-CNN model presented in Section 4. In the modified structure, physical parameters, i.e., the porosity (\(n^{F}\)) and the specific surface area (SSA), are incorporated directly into the network architecture instead of having \(n^{F}\) in the loss function. This results in a kind of physical parameter-informed CNN, which we call \"Informed \(K^{5}\)-CNN model\". The modified structure of the CNN model is illustrated in Figure 7.  
Specifically, an additional input branch (parallel branch) is added to the network to include \(n^{F}\) and SSA, which are passed through a dense layer with 16 units and _ReLU_ activation. Thus, the output of the Flatten layer (main branch) and the added parallel branch are concatenated to combine the information from both sources. Specifically, the concatenation layer merges the 1D feature vector from the image data and the supplementary features (\(n^{F}\) and SSA), allowing the model to exploit both image-based and additional data. As \(n^{F}\) is no longer part of the output data, a modified MSE loss function needs to be considered, which is expressed as  
\[\mathcal{D}_{\text{MSE}}=\frac{1}{n}\sum_{i=1}^{n}\left[\left(K^{5,p}_{kk,i}- K^{5,t}_{kk,i}\right)^{2}\right], \tag{30}\]  
where \(n\) is the number of output data points and \(k=1,2,3\).  
The training results of the \"Informed \(K^{5}\)-CNN model\" are evaluated through the loss vs. epoch curves, as illustrated in Figure 8. These show an initial rapid learning trend. However, similar to \([K^{5}-n^{F}]\)-CNN model, the validation loss begins to diverge from the training loss, indicating overfitting. Therefore, the model's state after approximately  
Figure 7: Illustration of the “Informed \(K^{5}\)-CNN model” architecture (see Figure 4 for comparison). This illustration shows the information flow and the inclusion of the physical information (\(n^{F}\) and SSA) through an additional input branch.  
100 epochs, corresponding to a loss value of around \(10^{-3}\), was considered optimal for further evaluation.  
The scatter plots in Figure 9, which compare the predicted values to the LBM-based ground truth values for the output variables \(K_{kk}^{S}\) (for \(k=1,2,3\)), further demonstrate the ability of the model to accurately predict the intrinsic permeability values.  
The performance of the \"Informed \(K^{S}\)-CNN model\" is quantitatively assessed using the \(R^{2}\) score.The model achieved an \(R^{2}\)-score of 0.983, which indicates a high level of accuracy. Although the structure of Model (2) is more complicated than that of Model (1), its \(R^{2}\) score is very close to that of Model (1). Thus, there is no significant advantage to using Model (2) over Model (1) in this particular model with the considered dataset. Based  
Figure 8: Model (2): Training and validation loss function values over the number of weight updates (Epoch). The validation loss (blue curve) initially follows a similar trend to that of training loss (red curve) but starts to diverge after approximately 100 epochs, suggesting overfitting beyond this point.  
Figure 9: Model (2): CNN predictions vs. ground truth values of the intrinsic permeability components. The close alignment of the predicted values with the red line indicates a high prediction accuracy, confirming the model’s effectiveness in estimating permeability and porosity values.  
on this, further research on other datasets is needed to explore the potential benefits of integrating additional physical parameters into neural network architectures.