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PDE-Free Mass-Constrained Learning of Complex Systems with Hidden States: The crowd dynamics case

Gianmaria Viola, Alessandro Della Pia, Lucia Russo, Ioannis Kevrekidis, Constantinos Siettos

TL;DR

This work tackles learning the dynamics of mass-conserving PDE-like systems with hidden closures by adopting a PDE-free framework that builds reduced-order models (ROMs) in a latent space. It combines delay-coordinate embeddings with either POD or Diffusion Maps to obtain latent coordinates, learns latent evolution via Multivariate Autoregressive modeling, and reconstructs high-dimensional fields through mass-preserving lifting (POD or k-NN). The authors prove mass conservation for POD and k-NN lifting and demonstrate superior performance of DM-informed ROMs over POD-based ROMs in the Hughes crowd model, achieving stable long-horizon predictions with significantly fewer latent coordinates. The approach offers a computationally efficient, non-intrusive pathway to capture complex, hidden dynamics in crowd and CFD-like systems, with potential broad applicability to PDEs where explicit operators are unknown or expensive to learn.

Abstract

We propose a machine learning framework based on the next-generation Equation-Free algorithm for learning the spatio-temporal dynamics of mass-constrained complex systems with hidden states, whose dynamics can in principle be described by PDEs, but lack explicit models. In these cases, some variables, closures, and potentials governing the dynamics are generally not directly observable and therefore must be inferred from data. Here, we construct manifold-ROMs -- using delayed coordinates, thus exploiting the Takens'/Whitney's embedding theorems. In the first stage, we employ both linear (POD) and nonlinear manifold learning (Diffusion Maps, DMs) to extract low-dimensional latent representations of the complex spatio-temporal evolution. In the second step, we learn predictive manifold-informed ROMs to approximate the solution operator on the latent space. In the final step, the latent dynamics are lifted back to the original high-dimensional space by solving a pre-image problem. We prove that both POD and the particular $k$-nearest neighbors lifting operators preserve the mass, a crucial property in the context of many problems, including computational fluid dynamics (CFD) and crowd dynamics. Actually, the proposed framework reconstructs the solution operator of the unavailable mass-constrained PDE, bypassing the need to discover an explicit form of the PDE per se. We demonstrate our approach via the Hughes model, approximating the dynamics of individuals minimizing travel time while avoiding obstacles and high-density regions. We show that DMs-informed ROMs outperform the best POD-informed ROMs thus resulting in stable and accurate approximations of the solution operator both in the latent space and, via reconstruction, in the high-dimensional space, and can therefore be integrated reliably over long time horizons.

PDE-Free Mass-Constrained Learning of Complex Systems with Hidden States: The crowd dynamics case

TL;DR

This work tackles learning the dynamics of mass-conserving PDE-like systems with hidden closures by adopting a PDE-free framework that builds reduced-order models (ROMs) in a latent space. It combines delay-coordinate embeddings with either POD or Diffusion Maps to obtain latent coordinates, learns latent evolution via Multivariate Autoregressive modeling, and reconstructs high-dimensional fields through mass-preserving lifting (POD or k-NN). The authors prove mass conservation for POD and k-NN lifting and demonstrate superior performance of DM-informed ROMs over POD-based ROMs in the Hughes crowd model, achieving stable long-horizon predictions with significantly fewer latent coordinates. The approach offers a computationally efficient, non-intrusive pathway to capture complex, hidden dynamics in crowd and CFD-like systems, with potential broad applicability to PDEs where explicit operators are unknown or expensive to learn.

Abstract

We propose a machine learning framework based on the next-generation Equation-Free algorithm for learning the spatio-temporal dynamics of mass-constrained complex systems with hidden states, whose dynamics can in principle be described by PDEs, but lack explicit models. In these cases, some variables, closures, and potentials governing the dynamics are generally not directly observable and therefore must be inferred from data. Here, we construct manifold-ROMs -- using delayed coordinates, thus exploiting the Takens'/Whitney's embedding theorems. In the first stage, we employ both linear (POD) and nonlinear manifold learning (Diffusion Maps, DMs) to extract low-dimensional latent representations of the complex spatio-temporal evolution. In the second step, we learn predictive manifold-informed ROMs to approximate the solution operator on the latent space. In the final step, the latent dynamics are lifted back to the original high-dimensional space by solving a pre-image problem. We prove that both POD and the particular -nearest neighbors lifting operators preserve the mass, a crucial property in the context of many problems, including computational fluid dynamics (CFD) and crowd dynamics. Actually, the proposed framework reconstructs the solution operator of the unavailable mass-constrained PDE, bypassing the need to discover an explicit form of the PDE per se. We demonstrate our approach via the Hughes model, approximating the dynamics of individuals minimizing travel time while avoiding obstacles and high-density regions. We show that DMs-informed ROMs outperform the best POD-informed ROMs thus resulting in stable and accurate approximations of the solution operator both in the latent space and, via reconstruction, in the high-dimensional space, and can therefore be integrated reliably over long time horizons.
Paper Structure (14 sections, 2 theorems, 50 equations, 7 figures, 4 tables)

This paper contains 14 sections, 2 theorems, 50 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Problem statement: Learning the dynamics of complex systems and the crowd dynamics paradigm
  3. The Methodology
  4. Manifold Learning and the Pre-image/Reconstruction Problem
  5. Encoding and Decoding with POD
  6. Decoding with POD conserves mass.
  7. Encoding with DMs and Decoding with the k-NN Algorithm
  8. Decoding with k-nearest neighbors
  9. Manifold-Informed Multivariate Autoregressive Modeling
  10. Assessing the Reconstruction Error Using the Wasserstein Distance
  11. The Case Study: Hughes Model for Crowd Dynamics
  12. Numerical Results
  13. Conclusions
  14. Numerical discretization of the Hughes model

Key Result

Proposition 1

Let the dataset be with columns $\mathbf{x}_k \in \mathbb{R}^{N}$ representing normalized to sum to 1 density fields. If the total mass of the original data is preserved, i.e., then the mass of the reconstructed/decoded field computed by POD as where $\mathbf{U}_d = [\mathbf{u}_1,\mathbf{u}_2,\dots, \mathbf{u}_d] \in \mathbb{R}^{N \times d}$ is the orthonormal basis formed by the first $d$ left

Figures (7)

  • Figure 1: Schematic of the PDE-free mass-preserving approach. (1) Manifold learning: projection of high-dimensional observations onto latent spaces; (2) Learning manifold-informed ROMs (i.e., learning the solution operator in the latent space) (here via Multivariate Autoregressive modeling (MVARs)); (3) Solution of the pre-image problem: construction of the map between the learned latent-space dynamics and the spatio-temporal dynamics (i.e., reconstruction of the solution operator of the unavailable PDE).
  • Figure 2: Schematic representation of the physical layout. A pedestrian crowd, initially distributed according to $\rho_0(x, y)$ (shown in red scale), moves through a rectangular domain $\Omega$ with a centrally placed obstacle $\Omega_{\text{obs}}$.
  • Figure 3: Contour maps of the Hughes PDE solution $\rho(t,x,y)$ for the Gaussian initial condition $\rho_0(x,y)$ (Eq. \ref{['eq:initial_condition']}) with $x_0=2.5$ m, $y_0=2.5$ m, $\sigma_x=1.8$ m, $\sigma_y=1.8$ m, at different time instants: (a) $t=0$ s; (b) $t=2$ s; (c) $t=5$ s; (d) $t=8$ s; (e) $t=13$ s; (f) $t=18$ s; (g) $t=20$ s; (h) $t=22$ s; (i) $t=28$ s; (j) $t=30$ s.
  • Figure 4: (a) DMs eigenvalues $\lambda_i$; (b) baseline mean reconstruction error $W_1$ computed over $d=4$, $7$, and $10$ DMs coordinates for the $N_{\text{train}} = 40$ training simulations; (c) cumulative distribution of POD eigenvalues (black circles) and 99% cumulative variance threshold (blue dashed line); (d) baseline mean reconstruction error $W_1$ for $d=18$, $20$, and $22$ POD coordinates on the training set.
  • Figure 5: Time evolution of reconstruction errors in the test set $X_{\text{test}}$. (a,b) Absolute $L_2$ error ($\varepsilon_2$); (c,d) relative $L_2$ error ($\varepsilon_2^r$); (e,f) Wasserstein distance ($W_1$) for the POD-informed ROM ($d=20,\,l=5$, left column) and DMs-informed ROM ($d=10,\,l=8$, right column) across the $N_{\text{test}}=40$ test simulations. In all panels, the black line denotes the mean error between the high-dimensional Hughes PDE solution (ground truth) and the ROM predictions, while shaded bands indicate the 10–90% percentiles.
  • ...and 2 more figures

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof