Deep learning-based models numerical solutions and their theoretical stability for a parabolic-parabolic chemotaxis models with nonlocal logistic sources in bounded heterogeneous environments

In this paper we solve numerically a parabolic-parabolic chemotaxis model with Lokta-Volterra type logistic sources chemotaxis in heterogeneous environments using Deep Neural Network (DNN) based models and study the convergence of numerical solutions to corresponding theoretical solutions and find a priori estimates of predictor error. In addition, we compare our deep learning-based model to the classical numerical methods and obtained similar results. However, the advantages of deep learning-based methods on numerical methods include solutions obtained that are not restricted to the grid points and we can predict the future dynamical behavior of the system.