Physics‐Informed Neural Networks for Solving the Keller–Segel Chemotaxis System

ABSTRACT The Keller–Segel model typically lacks an analytical solution, but its numerical solutions play a crucial role in biological modeling. Traditional methods, such as finite difference and finite element methods, face certain limitations when dealing with complex initial and boundary value problems. In contrast, Physics‐Informed Neural Networks (PINNs) offer a novel and effective approach for solving such problems. This paper applies the PINN method to solve the Keller–Segel model and demonstrates its excellent numerical performance. In Example 1, the Keller–Segel model is solved using both the finite difference method and PINNs. The results show that both methods are highly consistent, with small errors and well‐distributed patterns, highlighting the computational reliability of PINNs for this model. In Example 2, the Keller–Segel model with a flux source term is solved using PINNs, and the results are compared with the analytical solution. The high fitting accuracy further validates the precision of the method. Additionally, we compared the L 2 errors across different network architectures. Building on the validation from the first two examples, Example 3 uses the same PINN framework to solve a more complex one‐dimensional Keller–Segel model. Although this model lacks an analytical solution for reference, the solution exhibits structural and behavioral consistency, demonstrating the versatility and applicability of the PINN method. Our systematic exploration of 1D cases establishes theoretical foundations and practical guidelines for tackling high‐dimensional biophysical systems with complex boundaries, thereby advancing interdisciplinary applications of PINN‐based computational paradigms.