Linearized Implicit Methods Based on a Single-Layer Neural Network: Application to Keller-Segel Models

This paper is concerned with numerical approximation of some two-dimensional Keller-Segel chemotaxis models, especially those generating pattern formations. The numerical resolution of such nonlinear parabolic-parabolic or parabolic-elliptic systems of partial differential equations consumes a significant computational time when solved with fully implicit schemes. Standard linearized semi-implicit schemes, however, require reasonable computational time, but suffer from lack of accuracy. In this work, two methods based on a single-layer neural network are developed to build linearized implicit schemes: a basic one called the each step training linearized implicit (ESTLI) method and a more efficient one, the selected steps training linearized implicit (SSTLI) method. The proposed schemes make use also of a spatial finite volume method with a hybrid difference scheme approximation for convection-diffusion fluxes. Several numerical tests are performed to illustrate the accuracy, efficiency and robustness of the proposed methods. Generalization of the developed methods to other nonlinear partial differential equations is straightforward.