- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access
  • Access by Xinjiang University

Interpretability versus performance of analytical and neural-network-based permeability prediction models: Exploring separability, monotonicity, and dimensional consistency

Erik Jansson1,* and Magnus Röding1,2,3

  • *Contact author: erikjans@chalmers.se
PDF

Phys. Rev. E 111, 045509 – Published 25 April, 2025

DOI: https://doi.org/10.1103/PhysRevE.111.045509

Abstract

Effective mass transport properties of porous materials, such as permeability, are heavily influenced by their three-dimensional microstructure. There are numerous models developed for the prediction of permeability from microstructural characteristics, ranging from straightforward analytical relationships to high-performing machine learning models based on neural networks. There is an inherent tradeoff between predictive performance and interpretability; analytical models do not provide the best predictive performance but are relatively simple to understand. Neural networks, on the other hand, provide better predictive performance but are harder to interpret. In this paper, we investigate a multitude of models on the performance-versus-interpretability spectrum. Specifically, we use a dataset of 90000 microstructures developed elsewhere and consider the prediction of permeability using the microstructural descriptors porosity, specific surface area, and geodesic tortuosity. At the respective ends of the spectrum, we study analytical, power-law-type models and fully connected neural networks. In between, we study neural networks that are either separable, monotonic, or both separable and monotonic. Establishing monotonic relationships is particularly interesting considering the potential for solving the inverse microstructure design problem using gradient-based methods. In addition, we study versions of these models that are consistent and inconsistent in terms of physical dimension.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (55)

  1. S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer, New York, 2002).
  2. J. Kozeny, Sitzungsber. Akad. Wiss., Wien 136, 271 (1927).
  3. P. C. Carman, AIChE J. 15, 150 (1937).
  4. S. N. Chiu, D. Stoyan, W. Kendall, and J. Mecke, Stochastic Geometry and its Applications (Wiley, New York, 2013).
  5. D. Jeulin, Morphological Models of Random Structures (Springer, New York, 2021).
  6. L. Petrich, O. Furat, M. Wang, C. E. Krill, III, and V. Schmidt, Front. Mater. 8, 760602 (2021).
  7. B. Prifling, M. Ademmer, F. Single, O. Benevolenski, A. Hilger, M. Osenberg, I. Manke, and V. Schmidt, Comput. Mater. Sci. 192, 110354 (2021).
  8. O. Furat, L. Petrich, D. Finegan, D. Diercks, F. Usseglio-Viretta, K. Smith, and V. Schmidt, npj Comput. Mater. 7, 105 (2021).
  9. M. Neumann, J. Staněk, O. M. Pecho, L. Holzer, V. Beneš, and V. Schmidt, Comput. Mater. Sci. 118, 353 (2016).
  10. H. Moussaoui, J. Laurencin, Y. Gavet, G. Delette, M. Hubert, P. Cloetens, T. L. Bihan, and J. Debayle, Comput. Mater. Sci. 143, 262 (2018).
  11. M. Neumann, M. Osenberg, A. Hilger, D. Franzen, T. Turek, I. Manke, and V. Schmidt, Comput. Mater. Sci. 156, 325 (2019).
  12. P. Townsend, E. Larsson, T. Karlson, S. A. Hall, M. Lundman, P. Bergström, C. Hanson, N. Lorén, T. Gebäck, A. Särkkä, and M. Röding, Comput. Mater. Sci. 194, 110433 (2021).
  13. M. Weber, A. Grießer, D. Mosbach, E. Glatt, A. Wiegmann, and V. Schmidt, Sci. Rep. 13, 19359 (2023).
  14. S. Barman, H. Rootzén., and D. Bolin, AIChE J. 65, 446 (2019).
  15. B. Prifling, M. Neumann, D. Hlushkou, K. Kübel, U. Tallarek, and V. Schmidt, Comput. Mater. Sci. 187, 109934 (2021).
  16. D. Westhoff, J. Skibinski, O. Šedivý, B. Wysocki, T. Wejrzanowski, and V. Schmidt, Mater. Des. 147, 1 (2018).
  17. J. Rao, A. Püttmann, S. Khirevich, U. Tallarek, C. Geuzaine, M. Behr, and E. von Lieres, Comput. Chem. Eng. 178, 108355 (2023).
  18. L. Riley, P. Cheng, and T. Segura, Nat. Comput. Sci. 3, 975 (2023).
  19. A. Gayon-Lombardo, L. Mosser, N. P. Brandon, and S. J. Cooper, npj Comput. Mater. 6, 82 (2020).
  20. L. Xu, N. Hoffman, Z. Wang, and H. Xu, Mater. Des. 223, 111223 (2022).
  21. L. Holzer, P. Marmet, M. Fingerle, A. Wiegmann, M. Neumann, and V. Schmidt, Tortuosity and Microstructure Effects in Porous Media: Classical Theories, Empirical Data and Modern Methods (Springer, New York, 2023).
  22. M. Neumann, O. Stenzel, F. Willot, L. Holzer, and V. Schmidt, Int. J. Solids Struct. 184, 211 (2020).
  23. D. Hlushkou, H. Liasneuski, U. Tallarek, and S. Torquato, J. Appl. Phys. 118 (2015).
  24. Z. Ma and S. Torquato, Phys. Rev. E 98, 013307 (2018).
  25. L. Masaro and X. Zhu, Prog. Polym. Sci. 24, 731 (1999).
  26. J. H. van der Linden, G. A. Narsilio, and A. Tordesillas, Phys. Rev. E 94, 022904 (2016).
  27. O. Stenzel, O. Pecho, L. Holzer, M. Neumann, and V. Schmidt, AIChE J. 63, 4224 (2017).
  28. M. Röding, Z. Ma, and S. Torquato, Sci. Rep. 10, 15239 (2020).
  29. K. Graczyk and M. Matyka, Sci. Rep. 10, 21488 (2020).
  30. H. Wang, Y. Yin, X. Hui, J. Bai, and Z. Qu, Energy AI 2, 100035 (2020).
  31. S. Gärttner, F. Alpak, A. Meier, N. Ray, and F. Frank, Comput. Geosci. 27, 245 (2023).
  32. B. Prifling, M. Röding, P. Townsend, M. Neumann, and V. Schmidt, Front. Mater. 8, 786502 (2021).
  33. N. Archer and S. Wang, Decis. Sci. 24, 60 (1993).
  34. J. Sill and Y. Abu-Mostafa, Adv. Neural Inf. Process. Syst. 9, 634 (1996).
  35. J. Sill, Adv. Neural Inf. Process. Syst. 10, 661 (1997).
  36. H. Daniels and M. Velikova, IEEE Trans. Neural Netw. 21, 906 (2010).
  37. M. M. Fard, K. Canini, A. Cotter, J. Pfeifer, and M. Gupta, Adv. Neural Inf. Process. Syst. 29, 2919 (2016).
  38. S. You, D. Ding, K. Canini, J. Pfeifer, and M. Gupta, Adv. Neural Inf. Process. Syst. 30, 2981 (2017).
  39. A. Wehenkel and G. Louppe, Adv. Neural Inf. Process. Syst. 32, 1545 (2019).
  40. A. Gupta, N. Shukla, L. Marla, A. Kolbeinsson, and K. Yellepeddi, arXiv:1909.10662.
  41. X. Liu, X. Han, N. Zhang, and Q. Liu, Adv. Neural Inf. Process. Syst. 33, 15427 (2020).
  42. N. Nolte, O. Kitouni, and M. Williams, Expressive monotonic neural networks, arXiv:2307.07512.
  43. D. Runje and S. Shankaranarayana, Constrained monotonic neural networks, arXiv:2205.11775.
  44. K. Schladitz, J. Ohser, and W. Nagel, in 13th International Conference Discrete Geometry for Computer Imagery, edited by A. Kuba, L. Nyúl, and K. Palágyi (Springer, New York, 2007), pp. 247–258.
  45. M. Neumann, C. Hirsch, J. Staněk, V. Beneš, and V. Schmidt, Scand. J. Stat. 46, 848 (2019).
  46. T. Gebäck and A. Heintz, Commun. Comput. Phys. 15, 487 (2014).
  47. T. Gebäck, M. Marucci, C. Boissier, J. Arnehed, and A. Heintz, J. Phys. Chem. B 119, 5220 (2015).
  48. Q. Zou and X. He, Phys. Fluids 9, 1591 (1997).
  49. D.-A. Clevert, T. Unterthiner, and S. Hochreiter, in 4th International Conference on Learning Representations, ICLR 2016, edited by Y. Bengio and Y. LeCun (ICLR, San Juan, Puerto Rico, 2016).
  50. Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, in Neural Networks: Tricks of the Trade, edited by G. Montavon, G. B. Orr, and K.-R. Müller, 2nd ed. (Springer-Verlag, Berlin, 2012), pp. 9–48.
  51. X. Glorot and Y. Bengio, in JMLR W&CP: Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics (Proceedings of Machine Learning Research, Sardinia, Italy, 2010), Vol. 9, pp. 249–256.
  52. N. Qian, Neural Netw. 12, 145 (1999).
  53. L. Bottou, in Proceedings of COMPSTAT'2010 (Springer, New York, 2010), pp. 177–186.
  54. J. Ansel, E. Yang, H. He, N. Gimelshein, A. Jain, M. Voznesensky, B. Bao, P. Bell, D. Berard, E. Burovski, G. Chauhan, A. Chourdia, W. Constable, A. Desmaison, Z. DeVito, E. Ellison, W. Feng, J. Gong, M. Gschwind, B. Hirsh et al., in Proceedings of the 29th ACM International Conference on Architectural Support for Programming Languages and Operating Systems, ASPLOS'24 (Association for Computing Machinery, New York, NY, 2024), Vol. 2, pp. 929–947.
  55. M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg et al., TensorFlow: Large-scale machine learning on heterogeneous systems (2015), software available from https://www.tensorflow.org/.

Outline

Information

Phys. Rev. E 111, 045509– Published 25 April, 2025
  • Received 26 September 2024
  • Revised 6 February 2025
  • Accepted 18 March 2025
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation