Viola, Gianmaria (2026) Physics-Informed PDE and PDE-Free Learning of Crowd Dynamics via Machine Learning. [Tesi di dottorato]

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Abstract

Modelling complex multiscale systems whose governing equations are unknown, partially known, or affected by hidden non-local closures is a long-standing challenge in mathematics, physics and engineering. This thesis develops and systematically compares a hierarchy of three data-driven surrogate-modelling strategies—black-box, gray-box, and PDE-Free—for mass-conserving spatio-temporal dynamics, using continuum crowd dynamics as a prototypical benchmark. The reference high-fidelity model is the Hughes pedestrian-flow PDE, in which a conservation law is coupled to a hidden Eikonal potential encoding optimal travel-time strategies. Two main contributions are advanced. First, on the theoretical side, we prove that both Proper Orthogonal Decomposition (POD) and Diffusion Maps (DMs), combined with their natural lifting operators, conserve total mass exactly. This result provides a structural foundation for surrogate models that are compatible with conservation laws independently of the specific machine learning architecture employed. Second, on the methodological side, we develop and compare three surrogate frameworks of increasing degree of physical information: a physics-free (black-box) convolutional neural network learning the full discrete-time solution operator with soft mass conservation; a gray-box physics-informed machine learning scheme retaining the continuity equation as a hard constraint and learning the unknown velocity closure through a manifold-augmented neural network; and a PDE-Free framework where dynamics are learned and evolved directly in a low-dimensional latent space identified by manifold learning, with high-dimensional reconstruction performed on demand via mass-preserving lifting. Numerical experiments on the Hughes benchmark demonstrate that the proposed manifold-based surrogates achieve high accuracy, exact mass conservation, and a compact parametrisation across the three frameworks. These results indicate that, for mass-conserving complex systems with hidden states, manifold learning can substantially enhance the data-driven identification of macroscopic dynamics, with the PDE-Free framework achieving this through a particularly compact latent representation.

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