Next Generation Equation-Free Multiscale Modelling of Crowd Dynamics via Machine Learning

TL;DR

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 demonstrates high accuracy, robustness, and generalizability, thus allowing for fast and accurate modelling/simulation of crowd dynamics from agent-based simulations.

Abstract

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.

Figures (16)

• Figure 1: Schematic of the proposed methodology. First, microscopic distributions of pedestrian positions are mapped to macroscopic density fields (Step 1) and embedded into a latent space using POD (Step 2). Second, surrogate reduced-order models (MVARs and LSTMs) are trained in this latent space to capture and forecast the complex dynamics (Step 3). Finally, the learned dynamics are lifted back to the high-dimensional density fields via linear projection with the constructed POD basis (Step 4).
• Figure 2: Crowd dynamics configurations of SFM simulating pedestrians moving in a corridor past an obstacle. In the unidirectional flow configuration in Fig. \ref{['fig:Pedestrians']}(a), pedestrians move from the left to the right of the corridor. In the counterflow configuration in Fig. \ref{['fig:Pedestrians']}(b), two pedestrian populations move in opposite directions; group 1 (blue) moves from the left to the right, while group 2 (red) follows the opposite direction.
• Figure 3: Accuracy of the restriction and lifting operators in Eqs. \ref{['eq:Rest']} and \ref{['eq:Lift']} for the unidirectional flow case. Panel (a) shows the minimum number of POD modes (colored diamonds) required for retaining the desired percentage of energy in the training data set. Panel (b) depicts the average (over the $C=20$ cases with different initial conditions) reconstruction error $e^{2,rec}_k$ in Eq. \ref{['eq:PODreconErr']} with $d=6$ POD modes, in time for the training and testing datasets.
• Figure 4: Closed-loop/recursive simulation errors in the ambient--density profile--space, across the testing set over time, using the trained MVAR and LSTM models for the unidirectional flow case. Panels (a)--(d) display the relative $L_2$ (cyan) and $L_\infty$ (green) errors $e^{2,(c)}_k$ and $e^{\infty,(c)}_k$ in Eq. \ref{['eq:ForcastErr']} for MVAR(4), MVAR(9), LSTM(4) and LSTM(9) models, respectively. Panels (e) and (f) provide a system's "baseline" relative error estimation, obtained from the SFM simulations using $\pm1\%$ perturbations in the initial conditions for the density profiles (see in Section \ref{['sb:conf']}). Mean relative error (solid) and 10--90% percentiles (dashed) are shown over $C=20$ cases per time step. For the LSTM models, results correspond to the model achieving the best training performance out of the 50 independent runs.
• Figure 5: Closed-loop/recursive simulation errors in the ambient--density profile--space, across the testing set over time, using the MVAR and LSTM models for the counterflow case. Panels (a,b) and (c,d) display the relative reconstructed $L_2$ (cyan) and $L_\infty$ (green) errors $e^{2,(c)}_k$ and $e^{\infty,(c)}_k$ in Eq. \ref{['eq:ForcastErr']} for MVAR(10) and LSTM(10) models, respectively. The left and right column panels show errors for group 1 and 2, respectively. Mean relative errors (solid) and 10--90% error percentiles (dashed) are reported over $C=20$ cases per time step. For the LSTM model, results correspond to the run achieving the best training performance out of the 50 independent initializations.

Theorems & Definitions (7)

• Proposition 1
• proof
• Proposition 2
• proof
• Remark 2.1
• Remark 2.2
• Remark 3.1