Фiзично-iнформоване нейромережеве моделювання нелокальної динамiки натопу для сценарiїв евакуацiї

Анотація

We investigate a nonlocal continuum model of crowd dynamics using a physics-informed neural network approach. The crowd is described by a system of nonlinear conservation laws in which the flux incorporates advection, diffusion, and nonlocal interaction terms accounting for density-dependent motion and limited perception of surrounding agents. Nonlocal effects are modeled through spatial convolutions with smooth kernels, enabling agents to respond to averaged density gradients rather than purely local information. The governing system of partial differential equations is solved using a physics-informed neural network known as PINN, which approximates the solution over the entire space–time domain while enforcing the physical constraints through automatic differentiation. The nonlocal interaction terms are implemented in a stable discrete convolution form, ensuring numerical robustness during training. The approach is demonstrated on the interaction of two pedestrian groups moving in opposite directions in a one-dimensional corridor. The results exhibit the formation and propagation of density fronts, the gradual merging of flows, and the emergence of stable mixed zones. A characteristic feature of the solution is the partial interpenetration of the groups without rigid collisions, reflecting realistic collective motion. To validate the method, the PINN solution is compared with a reference finite-difference scheme based on a Rusanov flux. Qualitative agreement is observed in front structure and mixing dynamics, while quantitative deviations in key characteristics remain
within a few percent. A systematic parameter study shows that the PINN-based solution remains stable under variations of advection velocity, diffusion coefficient, and nonlocal interaction radius, in contrast to the finite-difference scheme, which exhibits strong stability limitations. These results demonstrate that PINN provides a robust and physically consistent tool for modeling nonlinear nonlocal crowd dynamics.