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

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.