Certifying Conservation Laws in Learned World Models

This research investigates a fundamental question in representation learning for physical world models: under what conditions do conservation laws remain certifiable after a model learns a latent representation? The study introduces "certified horizons," which are bounds that predict how many steps a model's rollout will provably adhere to a physical invariant's level set, based on measurable model defects. Crucially, the certification focuses on the decoded physical invariant, obtained by decoding the latent state and evaluating the known invariant, rather than relying on a learned latent Hamiltonian or scalar witness. The derived shell-horizon certificates decompose the budget into representation, readout, and latent-dynamics defects, with a bridge allowing a soft learned witness to yield a certified horizon for the decoded invariant. Experiments across various observation types (state, learned-lift, pixel) on conservative systems reveal that conservation certificates can indeed survive learned representations. However, the study highlights that not all geometric priors are equally robust; a controlled-Lipschitz-aligned soft invariant proved more resilient in learned-representation settings than hard canonical symplectic structures. This work emphasizes the importance of the decoded physical invariant as the central object for measuring and certifying robustness to representation learning.