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

Bridging the microscopic and the macroscopic modelling scales in crowd dynamics constitutes an important, open challenge for systematic numerical analysis, optimization, and control. We propose a combined manifold and machine learning approach to learn the discrete evolution operator for the emergent crowd dynamics in latent spaces from high-fidelity agent-based simulations. The proposed framework builds upon our previous works on next-generation Equation-free algorithms on learning surrogate models for high-dimensional and multiscale systems. Our approach is a four-stage one, 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 KDE. In the second step, based on manifold learning, we construct a map from the macroscopic ambient space into the latent space parametrized by a few coordinates based on POD of the corresponding density distribution. The third step involves learning reduced-order surrogate ROMs in the latent space using machine learning techniques, particularly LSTMs networks and MVARs. Finally, we reconstruct the crowd dynamics in the high-dimensional space in terms of macroscopic density profiles. We demonstrate that the POD reconstruction of the density distribution via SVD conserves the mass. With this "embed->learn in latent space->lift back to the ambient space" pipeline, we create an effective solution operator of the unavailable macroscopic PDE for the density evolution. For our illustrations, we use the Social Force Model to generate data in a corridor with an obstacle, imposing periodic boundary conditions. The numerical results demonstrate high accuracy, robustness, and generalizability, thus allowing for fast and accurate modelling/simulation of crowd dynamics from agent-based simulations.

Nonlinear Discrete-Time Observers with Physics-Informed Neural Networks

We use Physics-Informed Neural Networks (PINNs) to solve the discrete-time nonlinear observer state estimation problem. Integrated within a single-step exact observer linearization framework, the proposed PINN approach aims at learning a nonlinear state transformation map by solving a system of inhomogeneous functional equations. The performance of the proposed PINN approach is assessed via two illustrative case studies for which the observer linearizing transformation map can be derived analytically. We also perform an uncertainty quantification analysis for the proposed PINN scheme and we compare it with conventional power-series numerical implementations, which rely on the computation of a power series solution.

Discrete-Time Nonlinear Feedback Linearization via Physics-Informed Machine Learning

We present a physics-informed machine learning (PIML) scheme for the feedback linearization of nonlinear discrete-time dynamical systems. The PIML finds the nonlinear transformation law, thus ensuring stability via pole placement, in one step. In order to facilitate convergence in the presence of steep gradients in the nonlinear transformation law, we address a greedy-wise training procedure. We assess the performance of the proposed PIML approach via a benchmark nonlinear discrete map for which the feedback linearization transformation law can be derived analytically; the example is characterized by steep gradients, due to the presence of singularities, in the domain of interest. We show that the proposed PIML outperforms, in terms of numerical approximation accuracy, the traditional numerical implementation, which involves the construction--and the solution in terms of the coefficients of a power-series expansion--of a system of homological equations as well as the implementation of the PIML in the entire domain, thus highlighting the importance of continuation techniques in the training procedure of PIML.