# A Multiscale CNN-Based Intrinsic Permeability Prediction in Deformable Porous Media
## 8 Conclusions and Future Aspects  
In this work, we have demonstrated the ability of different CNN models to accurately predict the anisotropic intrinsic permeability tensor together with the porosity at different deformation states. The inputs of this CNN model are real CT images related to Bentheim sandstone. The predicted parameters (outputs) can be directly integrated into a macroscopic TPM framework to describe both linear and non-linear flow through porous media, where different flow models have been discussed. Three CNN models are presented in this study: (1) an initial model that includes the permeability components and porosity in the definition of the loss function; (2) an informed CNN model that incorporates physical parameters directly into the model architecture; and (3) an enriched CNN model with transfer learning, where learning starts with synthetic data and the pre-trained model is fine-tuned on the original dataset. The initial CNN model showed high accuracy, while the informed CNN model did not significantly outperform the simpler initial model on the given dataset. However, data augmentation with synthetic data and the application of transfer learning proved effective in improving learning efficiency.  
Future research will focus on several promising areas to further improve and extend current models. This includes exploring the potential of using CNN models for the inverse design of porous metamaterials with tailored hydro-mechanical properties, such as stiffness and permeability, which can be fabricated by additive manufacturing. Another future approach is physics-enhanced multiscale modeling, where physics-based constraints and principles are integrated into the multiscale modeling framework. This approach could potentially allow for the use of smaller datasets, improving the efficiency and feasibility of training accurate models without compromising performance. While the proposed model effectively predicts anisotropic permeability, future studies could test its application to fractured porous media, where anisotropic flow dominates. Regarding the model's potential applicability to porous media beyond Bentheim sandstone, several approaches can be explored. For instance, transfer learning can be utilized, where the initial training is conducted on Bentheim sandstone and then fine-tuned for other materials. Additionally, physics-constrained models could be incorporated, as they are known to enhance accuracy while requiring fewer data. However, these approaches would necessitate further validation in future works to ensure robustness and reliability.  
Conceptualization, Y.H.; Methodology, Y.H., F.A. and W.E.; Software, Y.H.; Resources, F.A.; Writing--original draft, Y.H. and F.A.; Supervision, W.E. All authors have read and agreed to the published version of the manuscript.  
[PERSON] would like to gratefully thank the German Research Foundation (DFG) for its funding support through the project \"Multi-field continuum modeling of two-fluid-filled porous media fracture augmented by microscale-based machine-learning material laws\", grant number 458375627. [PERSON] gratefully acknowledges funding support for this research by the \"German Research Foundation\" (DFG) through the SFB/TRR-298-SIIRI--Project-ID 426335750.  
Not applicable.  
**Data Availability Statement:**: Exemplary source codes are available as open access for interested readers at [[https://doi.org/10.25835/xrii0m6f](https://doi.org/10.25835/xrii0m6f)]([https://doi.org/10.25835/xrii0m6f](https://doi.org/10.25835/xrii0m6f)) (accessed on 1 January 2025).  
The authors declare no conflicts of interest.