STDDN: A Physics-Guided Deep Learning Framework for Crowd Simulation

Abstract

Accurate crowd simulation is crucial for public safety management, emergency evacuation planning, and intelligent transportation systems. However, existing methods, which typically model crowds as a collection of independent individual trajectories, are limited in their ability to capture macroscopic physical laws. This microscopic approach often leads to error accumulation and compromises simulation stability. Furthermore, deep learning-driven methods tend to suffer from low inference efficiency and high computational overhead, making them impractical for large-scale, efficient simulations. To address these challenges, we propose the Spatio-Temporal Decoupled Differential Equation Network (STDDN), a novel framework that guides microscopic trajectory prediction with macroscopic physics. We innovatively introduce the continuity equation from fluid dynamics as a strong physical constraint. A Neural Ordinary Differential Equation (Neural ODE) is employed to model the macroscopic density evolution driven by individual movements, thereby physically regularizing the microscopic trajectory prediction model. We design a density-velocity coupled dynamic graph learning module to formulate the derivative of the density field within the Neural ODE, effectively mitigating error accumulation. We also propose a differentiable density mapping module to eliminate discontinuous gradients caused by discretization and introduce a cross-grid detection module to accurately model the impact of individual cross-grid movements on local density changes. The proposed STDDN method has demonstrated significantly superior simulation performance compared to state-of-the-art methods on long-term tasks across four real-world datasets, as well as a major reduction in inference latency.

Figures (10)

• Figure 1: Overview of the STDDN framework, consisting of a microscopic trajectory prediction network (bottom left), a neural ODE-based macroscopic density evolution module (top left), and the DVCG module (right), which connects microscopic trajectories with macroscopic density and velocity fields. The DVCG module includes three components—DDM, CGD, and NE—combined in a dynamic Graph Neural Network (GNN) to model density evolution. These components build the inflow term $\mathbf{G}_{in}(\Phi, t, \rho^{t})$ and outflow term $\mathbf{G}_{out}(\Phi, t, \rho^{t})$, which are combined via Eq.\ref{['eq4']} to form the derivatives of the density field. Colored arrows show velocity vectors at different time steps.
• Figure 2: Accumulate error of the simulation, using MAE and OT as metrics.
• Figure 3: Sensitivity analysis on the GC dataset for grid size, ODE time steps $\tau$, loss balancing coefficients ($\lambda_1$ and $\lambda_2$), and node embedding dimension, using MAE as the evaluation metric.
• Figure 4: The detailed of next trajectory prediction model.
• Figure 5: Visualization of predicted trajectories on the GC, UCY, ETH, and HOTEL datasets. Each column represents a different dataset, showcasing various scenarios. Observed trajectories are shown in , ground truth in , STDDN in , SPDiff in , NSP in , and PCS in .