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 method and a more efficient one, the selected steps training linearized implicit method. The proposed schemes, which make use also of a spatial finite volume method with a hybrid difference scheme approximation for convection–diffusion fluxes, are first derived for a chemotaxis system arising in embryology. The convergence of the numerical solutions to a corresponding weak solution of the studied system is established. Then the proposed methods are applied to a number of chemotaxis models, and several numerical tests are performed to illustrate their accuracy, efficiency and robustness. Generalization of the developed methods to other nonlinear partial differential equations is straightforward.

本文涉及一些二维Keller-Segel化学趋化模型的数值逼近，尤其是那些产生图案形成的模型。当使用完全隐式格式求解时，这种非线性的抛物型-抛物型或抛物型-椭圆型方程组的数值分辨率会消耗大量的计算时间。但是，标准的线性半隐式方案需要合理的计算时间，但缺乏准确性。在这项工作中，开发了两种基于单层神经网络的方法来构建线性化隐式方案：一种基本的方法称为“每步训练线性化隐式方法”，一种更有效的方法是“选择的步骤”训练线性化隐式方法。拟议的计划，它首先利用对流扩散流的混合差分方案近似的空间有限体积方法，首先针对胚胎学中产生的趋化性系统推导。建立了数值解与研究系统相应弱解的收敛性。然后将所提出的方法应用于许多趋化性模型，并进行了一些数值测试，以说明其准确性，效率和鲁棒性。将开发的方法推广到其他非线性偏微分方程很简单。然后将所提出的方法应用于许多趋化性模型，并进行了一些数值测试以说明其准确性，效率和鲁棒性。将开发的方法推广到其他非线性偏微分方程很简单。然后将所提出的方法应用于许多趋化性模型，并进行了一些数值测试以说明其准确性，效率和鲁棒性。将开发的方法推广到其他非线性偏微分方程很简单。