This article is devoted to developing a deep learning method for the numerical solution of the partial differential equations (PDEs). Graph kernel neural networks (GKNN) approach to embedding graphs into a computationally numerical format has been used. In particular, for investigation mathematical models of the dynamical system of cancer cell invasion in inhomogeneous areas of human tissues have been considered. Neural operators were initially proposed to model the differential operator of PDEs. The GKNN mapping features between input data to the PDEs and their solutions have been constructed. The boundary integral method in combination with Green’s functions for a large number of boundary conditions is used. The tools applied in this development are based on the Fourier neural operators (FNOs), graph theory, theory elasticity, and singular integral equations.
References
[1]
Chumburidze, M. and Niminet, V. (2025) Efficient Modelisation for Complex Geometries of Tumor Growth via Thermo-Elastic Diffusion Partial Differential Equations and Artificial Neural Networks. Broad Research in Artificial Intelligence and Neuroscience, 16, 315-323. https://doi.org/10.70594/brain/16.1/23
[2]
Climescu-Haulica, A. (2024) The Brain’s Essential Role in Mediating Immune Responses: HPA Axis to Leverage Signals with a Systemic Approach. Broad Research in Artificial Intelligence and Neuroscience, 15, 375-386. https://doi.org/10.70594/brain/15.4/25
[3]
Boullé, N., Earls, C.J. and Townsend, A. (2022) Data-Driven Discovery of Green’s Functions with Human-Understandable Deep Learning. Scientific Reports, 12, Article No. 4824. https://doi.org/10.1038/s41598-022-08745-5
[4]
Chumburidze, M. and Lekveishvili, D. (2017) Numerical Approximation of Basic Boundary-Contact Problems. ASME 2017 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Cleveland, 6-9 August 2017, DETC2017-67097.
[5]
Chumburidze, M. (2016) Approximate Solution of Some Boundary Value Problems of Coupled Thermo-elasticity. In: Bélair, J., Frigaard, I., Kunze, H., Makarov, R., Melnik, R., Spiteri, R., Eds., Mathematical and Computational Approaches in Advancing Modern Science and Engineering, Springer International Publishing, 71-80. https://doi.org/10.1007/978-3-319-30379-6_7
[6]
Goswami, S., Bora, A., Yu, Y. and Karniadakis, G.E. (2023) Physics-informed Deep Neural Operator Networks. In: Rabczuk, T. and Bathe, K.J., Eds., Computational Methods in Engineering & the Sciences, Springer International Publishing, 219-254. https://doi.org/10.1007/978-3-031-36644-4_6
[7]
Lanthaler, S., Mishra, S. and Karniadakis, G.E. (2022) Error Estimates for DeepONets: A Deep Learning Framework in Infinite Dimensions. Transactions of Mathematics and Its Applications, 6, tnac001. https://doi.org/10.1093/imatrm/tnac001
[8]
Li, Z., Kovachki, N., Azizzadenesheli, K., Liu, B., Bhattacharya, K., Stuart, A. and Anandkumar, A. (2020) Neural Operator: Graph Kernel Network for Partial Differential Equations. arXiv: 2003.03485. https://doi.org/10.48550/arXiv.2003.03485
[9]
Zhang, E., Kahana, A., Turkel, E., Ranade, R., Pathak, J. and Karniadakis, G.E. (2022) A Hybrid Iterative Numerical Transferable Solver (HINTS) for PDEs Based on Deep Operator Network and Relaxation Methods. arXiv: 2208.13273.
[10]
Chumburidze, M. and Lekveishvili, D. (2023) Approximation Solution of Mixed Boundary-Contact Problems and Their Applications. EQUATIONS, 3, 140-144. https://doi.org/10.37394/232021.2023.3.17
[11]
Cosentino Lagomarsino, M., Jona, P., Bassetti, B. and Isambert, H. (2007) Hierarchy and Feedback in the Evolution of the Escherichia coli Transcription Network. Proceedings of the National Academy of Sciences, 104, 5516-5520. https://doi.org/10.1073/pnas.0609023104
