Gradient-enhanced physics-informed neural networks for numerical approximation and parameter estimation of reaction–diffusion systems with chemotaxis and nonlocal terms

Purpose This study aims to explore a mathematical model of chemotaxis, the directed movement of cells or microorganisms in response to chemical gradients, by using a novel, meshfree approach. Specifically, it investigates the effectiveness of gradient-enhanced physics-informed neural networks (gPINNs) in solving this model, with the goal of improving precision and computational efficiency. Design/methodology/approach The research introduces a coupled system of partial differential equations (PDEs) that describes the chemotaxis process, incorporating a nonlocal integral term. To solve this system, the study uses the gPINNs method. Several numerical experiments are conducted to assess the performance of gPINNs, comparing its results against the standard physics-informed neural networks (PINNs) and the generalized finite difference method (GFDM). Findings The study demonstrates that gPINNs offer improved accuracy and efficiency compared to traditional PINNs and GFDM in solving the chemotaxis model. Two cases of parabolic-elliptic models, derived from the parabolic–parabolic system, are also examined. These cases illustrate the feasibility of discretizing nonlocal variables, validating the effectiveness of the proposed method in handling complex models. Originality/value This paper introduces a novel approach to solving the chemotaxis problem by applying gPINNs, which can be a valuable tool for modeling and simulating biological processes governed by complex PDE systems. The results contribute to the growing body of research on meshfree methods for solving PDEs and offer a new pathway for parameter estimation in these types of models.