PDE-Free Mass-Constrained Learning of Complex Systems with Hidden States

Abstract

We propose a three-tier 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 the first step, we employ Diffusion Maps (DMs), a nonlinear manifold learning algorithm, to extract low-dimensional latent representations of the complex spatio-temporal evolution. In the second step, we learn manifold-informed reduced-order models (ROMs) with Sparse Identification of Nonlinear Dynamics (SINDy) and standard linear Multivariate Autoregressive models (MVARs) 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 an (ill-posed) pre-image problem via a convex interpolation based on the k-NN algorithm. In doing so, the proposed framework reconstructs the solution operator of the unknown mass-constrained PDE, without explicitly identifying the PDE itself. For comparison purposes, we also evaluated the performance of the scheme for constructing ROMs based on Proper Orthogonal Decomposition (POD) and prove that both POD and the k-NN lifting operators preserve the mass. We illustrate the approach using two benchmark problems: (a) the Hughes model of crowd dynamics, which minimizes walking time while avoiding obstacles and high-density regions, and (b) a CFD problem involving the spatio-temporal evolution of a passive tracer advected by a periodic Navier--Stokes velocity field. We show that ROMs informed by DMs yield parsimonious models that consistently outperform the best POD-informed ROMs, yielding stable and accurate approximations of the solution operator in the latent space and, via reconstruction, in the original high-dimensional space over long time horizons.