用机器学习建模人群密度演化，实现快速高精度仿真。

Bridging the microscopic and macroscopic modelling scales in crowd dynamics constitutes an open challenge for systematic numerical analysis, optimization, and control. Here, we propose a manifold-informed machine learning approach to learn the discrete evolution operator for the emergent/collective crowd dynamics in latent spaces from high-fidelity individual/agent-based simulations. The proposed framework is a four-stage one, \textit{explicitly conserving the mass} of the reconstructed dynamics in the high-dimensional space. In the first step, we derive continuous macroscopic fields (densities) from discrete microscopic data (pedestrians' positions) using Kernel Density Estimation. In the second step, we construct a map from the density-field space into an appropriate latent space parametrized by a few coordinates based on Proper-Orthogonal Decomposition (POD) of the corresponding density distributions. The third step involves learning reduced-order surrogate models in the latent space using machine learning techniques, particularly Long Short-Term Memory networks and Multivariate Autoregressive models. Finally, we reconstruct the crowd dynamics in the high-dimensional space with POD, demonstrating that the POD reconstruction conserves the mass. Thus, with this ``embed -> learn in latent space -> lift back to the high-dimensional space'' pipeline, we create an effective solution operator of the unavailable (at the macroscopic scale) PDE for the evolution of the density distribution. For our illustrations, we used the Social Force Model to generate data in a corridor with an obstacle, imposing periodic boundary conditions in two scenarios: (i) a unidirectional flow, and (ii) a counterflow. The numerical results demonstrate high accuracy, robustness, and generalizability, thus allowing for fast and accurate modelling/simulation of crowd dynamics from agent-based simulations.