Beyond Reconstruction: Dynamical Admissibility in Physics-Based Temporal Modeling

Learning representations from observations supports physics-based modeling of evolving systems. Accurate reconstruction and small mechanistic residuals on sampled trajectories do not, by themselves, establish whether a learned representation supports closed latent evolution. We study the dynamical admissibility of time-varying representations, connecting temporal causal representation learning with dynamical systems. Using semiconjugacy and exact lumpability, we distinguish latent closure from compatibility with a prescribed mechanistic flow. Invertible reparameterizations preserve closure through conjugate dynamics, whereas retaining the prescribed flow requires them to commute with that flow. In approximate settings, we separate representation-induced nonclosure from transition-model error, and derive finite-horizon bounds relating nonclosure to reconstruction that identify when accurate reconstruction controls nonclosure and when unresolved future-state information imposes a tradeoff between future reconstruction accuracy and latent closure. Experiments on controlled synthetic systems and real-world cerebral perfusion data illustrate these relationships and the complementary roles of reconstruction accuracy, latent-transition consistency, and mechanistic consistency.