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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.

has been cited by the following article:

• TITLE: Numerical Study of Dynamical System Using Deep Learning Approach

  AUTHORS: Manana Chumburidze, Miranda Mnatsakaniani, David Lekveishvili, Nana Julakidze

  KEYWORDS: Deep Learning, Graph Kernel Network, Green’s Tensor

  JOURNAL NAME: Open Journal of Applied Sciences, Vol.15 No.2, February 21, 2025

  ABSTRACT: 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.