Distance-informed Neural Eikonal Solver for reactive dynamic user-equilibrium of macroscopic continuum traffic flow model

This paper revisits the Reactive Dynamic User-Equilibrium (RDUE) model for dynamic traffic assignment (DTA) of macroscopic traffic flow in two-dimensional continuum space, focusing on the Eikonal equation—a crucial partial differential equation (PDE) with specific boundary conditions. Traditionally, solving Eikonal equations has relied on iterative numerical methods through the discretization of the continuum space. However, this discretization compromises the precision of numerical solutions and could lead to non-convergence issues during iterative processes. This study refers to Physics-Informed Neural Networks (PINNs) and develops the Distance-Informed Neural Eikonal Solver (NES-DI) for solving Reactive Dynamic User-Equilibrium models. While the previously proposed Neural Eikonal Solver (NES) performs badly in a strong heterogeneous cost field with large cost differences, NES-DI explicitly considers the influence of solid boundaries during the factorization process by incorporating accurate distance information. Numerical examples of RDUE at both the static and dynamic levels are presented to illustrate the performance and applications of the NES-DI framework. The results demonstrate that NES-DI greatly outperforms both NES and the fast sweeping method. Moreover, NES-DI overcomes the limitations of discretization, enabling predictions of solutions at arbitrary locations within the computational domain. At the dynamic level, transfer learning is employed to leverage historical solutions to solve RDUE problems more efficiently. Overall, NES-DI shows the potential of solving reactive dynamic problems with strong heterogeneity, which offers a promising alternative to discretization-reliant numerical methods.