Reduced-order modeling (ROM) of time-dependent partial differential equations enables efficient simulation and inference of high-dimensional physical systems by learning dynamics on low-dimensional latent manifolds. In this talk we introduce interpretable Φ-ROM, a reduced-order modeling framework that integrates a differentiable physics solver directly into the training loop to enforce governing equations in latent space. Latent dynamics are learned via sparse regression of latent time derivatives over a library of candidate latent operators, yielding closed-form, physically interpretable evolution equations amenable to analysis and discovery. Beyond interpretability, the proposed framework addresses the simulation-to-data gap by enabling data assimilation from sparse and irregular observations. An implicit neural representation decoder reconstructs full solution fields from limited sensor data while constraining latent evolution through solver-informed sparse dynamics. This formulation ensures consistency with the underlying discretized physics while uncovering latent governing structures that explain discrepancies between simulated and observed dynamics.