In this paper, a mathematical model of moisture transfer in capillary-porous media in one- and two-dimensional space is presented and investigated, for its description the apparatus of fractional integrodifferentiation was used. This approach made it possible to take into account such properties of a system with a fractal structure as memory, self-organization, temporal and spatial correlations. The complexity of this mathematical model complicates its application and requires significant computing power. To calculate the numerical solution of the differential equation and speed up the calculations, the fractal neural network method, which is based on the fPINN architecture, is used. This method uses a loss function that takes into account physical information about the process under study. Formulas from fractional differential calculus were applied to express fractional derivatives and a difference scheme for the loss function was developed. The software for the implementation of the neural network method was developed and the applied approach was justified by comparing the obtained numerical results with the results of experiments by other scientists and the results obtained using finite difference numerical methods. The reliability check of the investigated indicators indicates the adequacy of the mathematical model and the prospects for further application of the numerical fractal neural network method.

1. V. V. Uchaikin, "Method drobovykh pokhidnykh." ["Method of Fractional Derivatives"]. Ul'yanovsk: Artishok Publ., 2008, 512 p.

2. L. A. Fil'shtyns'kyi, T. V. Mukomel, T. A. Kirichok. "Odnovymirna pochatkovo –kraiova zadacha dlia drobovo – diferentsiial'noho rivniannia teplonosinnia." [One-Dimensional Initial and Boundary Problem for Fractional Differential Equation of Heat Conduction"]. Visnyk Zaporiz'koho natsional'noho universytetu, 2010, No. 1, pp. 113- 118.

3. V. V. Nikolenko, V. A. Yachmenyev, "Tochne virіshennya pochatkovo-kraiovoji zadachi dlya rivnannya anormal'noji dyfuziyi." ["Exact Solution of the Initial-Boundary Problem for the Equation of Anomalous Diffusion"]. Visnyk Kharkivs'koho natsional'noho universytetu imeni V.M. Karazina, 2015, S. 131-136.

4. F. Huang, F.Liu, “The Space-Time Fractional Diffusion Equation with Caputo Derivatives”. Journal of Applied Mathematics and Computing, 19 (2005), 1-2, pp. 179-190. https://doi.org/10.1007/BF02935797

5. Y. Z Povstenko, «Signalling problem for time-fractional diffusion-wave equation in a half-plane». Fractional calculus and applied analysis, 2008, 11, № 3, pp. 329-352.

6. B.Y. Datsko, V.V. Gafiychuk, “Different types of instabilities and complex dynamics in reaction-diffusion systems with fractional derivatives”. Computational and Nonlinear Dynamics, 2012. https://doi.org/10.1115/1.4005923

7. Y.Povstenko, “Fundamental solutions to timefractional heat conduction equations in two joint half-lines”, Cent. Eur. J. Fhys, 11(10), 2013, рp. 1284-1294. https://doi.org/10.2478/s11534-013-0272-7

8. Zhu Tieyuan, Jerry M Harris, “Modeling acoustic wave propagation in heterogeneous attenuating media using decoupled fractional laplacians”, Geophysics, 79(3), 2014, T105–T116. https://doi.org/10.1190/geo2013-0245.1

9. Chinenye Assumpta Nnakenyi, J. A. C. Weideman, “Spectral Methods for Fractional Differential Equations”. African Institute for Mathematical Sciences (AIMS), 2015.

10. Ya. Sokolovskyy, V. Shymanskyi, M. Levkovych, “Mathematical modeling of nonisotermal moisture transfer and visco-elastic deformation in the materials with fractal structure”, Computer Science and Information Technologies. CSIT 2016: proc. of the 11th Intern. Sci. and Techn. Conf., Lviv, 2016. P. 91-95. https://doi.org/10.1109/STC-CSIT.2016.7589877

11. Y. Sokolovskyy, M. Levkovych, V. Yarkun, Y. Protsyk and A. Nechepurenko, "Parallel Realization of the Problem of Heat and Moisture Transfer in Fractal-structure Materials," 2020 IEEE 15th International Conference on Computer Sciences and Information Technologies (CSIT), Zbarazh, Ukraine, 2020, pp. 86-90, doi: 10.1109/CSIT49958.2020.9322021.

12. A. K. Bazzaev, "Lokalno-odnomirna skhema dlya rivnannya teplorozvidnosti iz krayovymi umovami tret'oho rodu." ["Local One-Dimensional Scheme for the Heat Conduction Equation with Third-Kind Boundary Conditions"]. Vladikavkazkyi matematychnyi zhurnal, 2011, Vol. 13, No. 1, pp. 3-12.

13. I.E. Lagaris; A. Likas; D.I. Fotiadis, “Artificial neural networks for solving ordinary and partial differential equations”. IEEE transactions on neural networks 1998, 9, 987–1000. https://doi.org/10.1109/72.712178

14. Malek, A.; Beidokhti, R.S. Numerical solution for high order differential equations using a hybrid neural network-optimization method. Applied Mathematics and Computation, 2006, 183, 260–271. https://doi.org/10.1016/j.amc.2006.05.068

15. J. Berg, K. Nystrom, “A unique deep artificial neural network approach to partial differential equations in complex geometries”. Neurocomputing, 317:28, https://doi.org/10.1016/j.neucom.2018.06.056

16. A. G. Baydin, B. A. Pearlmutter, A. A. Radul, J. M. Siskind. “Automatic differentiation in machine learning: a survey”. The Journal of Machine Learning Research, 18(1), 2017.

17. James Bergstra, Olivier Breuleux, Frédéric Bastien, Pascal Lamblin, Razvan Pascanu, Guillaume Desjardins, Joseph Turian, David Warde-Farley, Yoshua Bengio. “Theano: a cpu and gpu math expression compiler”. In Proceedings of the Python for scientific computing conference (SciPy), pages 1- 7, 2010. https://doi.org/10.25080/Majora-92bf1922-003

18. Yu M Ermoliev and RJ-B Wets, “Numerical techniques for stochastic optimization”. Springer-Verlag,1988. https://doi.org/10.1007/978-3-642-61370-8

19. I. E. Lagaris, A. Likas, D. I. Fotiadis. “Artificial Neural Networks for Solving Ordinary and Partial Diferential Equations”. arXiv e-prints, page physics/9705023, 1997.

20. P. Márquez-Neila, M. Salzmann, P. Fua, “Imposing Hard Constraints on Deep Networks: Promises and Limitations”. arXiv e-prints, page arXiv:1706.02025, 2017.

21. R. Kondor, S. Trivedi, “On the Generalization of Equivariance and Convolution in Neural Networks to the Action of Compact Groups”. Proceedings of the 35th International Conference on Machine Learning, Proceedings of Machine Learning Research, vol 80. PMLR, pp 2747-2755.

22. S. Mallat, “Understanding deep convolutional networks”. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2015, pp.203. https://doi.org/10.1098/rsta.2015.0203.

23. M. Raissi, P. Perdikaris, GE. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations”. Journal of Computational Physics, 378: 686-707. https://doi.org/10.1016/j.jcp.2018.10.045.

24. G. Pang, L. Lu, G. E. Karniadakis, “fPINNs: Fractional physics-informed neural networks”. SIAM Journal on Scientific Computing, 41, 2019, pp. A2603–A2626.

25. J. Bangti, L. Raytcho, Zhou Zhi. “An analysis of the l1 scheme for the subdiffusion equation with nonsmooth data”. IMA Journal of Numerical Analysis, 36(1), 2015, pp.197–221.

26. Y. Sokolovskyy, V. Shymanskyi, M. Levkovych, V. Yarkun, “Mathematical modeling of heat and moisture transfer and reological behavior in materials with fractal structure using the parallelization of predictor-corrector numerical method.” Proceedings of the 2016 IEEE 1st International Conference on Data Stream Mining and Processing, DSMP 2016, pp. 108-111, doi: 10.1109/DSMP.2016.7583518

27. Y. Sokolovskyy, M. Levkovych, O. Mokrytska, Y. Kaspryshyn, “Mathematical Modeling of Nonequilibrium Physical Processes, Taking into Account the Memory Effects and Spatial Correlation,” 9th International Conference on Advanced Computer Information Technologies, ACIT 2019 – Proceedings, 2019, pp. 56-59, doi: 10.1109/ACITT.2019.8780011