Data-driven methods (including reduced order models, machine-learning surrogates, or data-assimilation techniques) for physics-based simulations have demonstrated remarkable utility and effectiveness; however, achieving robust performance in extrapolatory and predictive regimes often remains challenging. Embedding physical structure-preserving properties (e.g., balance laws, preservation of geometric or solution invariants) can significantly improve performance in these regimes. These techniques also improve performance in the presence of challenging fluid physics, including shock waves, boundary layers, and turbulence. This minisymposium focuses on techniques for enforcing such properties within data-driven methods, which can both generalize and deviate significantly from approaches used in traditional numerical methods.