🤖 AI Summary
This study investigates whether physical conservation laws can be rigorously certified in learned latent representations and quantifies the temporal horizon over which such conservation remains valid in roll-out predictions. To this end, the authors introduce the notion of a "certified time horizon" and develop a certification framework centered on decoding physical invariants. Prediction error is decomposed into contributions from representation, readout, and latent dynamics. A monotonic alignment between soft and hard certification is established via shell-like temporal certificates and soft witness functions. The approach integrates Lipschitz control, geometric priors (e.g., symplectic structure), and invariant decoding, and is validated across state-space, learned coordinates, and pixel-level observations. Experiments show that hard symplectic constraints excel in known phase spaces but lack generalizability to learned coordinates, whereas soft alignment proves more robust in learned representations; the Kepler problem delineates the limits of geometric constraints, and pixel-level certification is recoverable within stable readout sub-tubes.
📝 Abstract
We ask a representation-learning question about physical world models: when does a conservation law remain certifiable after a model learns a latent representation? A certified horizon bounds -- in advance, from measurable model defects -- how many steps a rollout provably stays on a physical invariant's level set. The key design choice is what is certified: not a learned latent Hamiltonian or a learned scalar witness (a model can conserve either while drifting in true energy), but the decoded physical invariant obtained by decoding the latent state and evaluating the known invariant. Around this object we derive shell-horizon certificates whose budget decomposes into representation, readout, and latent-dynamics defects, with a monotone alignment bridge through which a soft learned witness yields a certified horizon for the decoded invariant, and test them across state, learned-lift, and pixel observations on conservative systems. Conservation certificates can survive learned representation, but not all geometric priors survive equally: hard canonical symplectic structure yields the longest horizons in known phase coordinates yet does not cross a learned chart, whereas a controlled-Lipschitz-aligned soft invariant survives in the learned-representation settings we test; pixel certification is recovered on a readout-stable sub-tube; and the Kepler problem exposes a geometric boundary. The central object is therefore not a latent Hamiltonian, but a decoded physical invariant whose robustness to representation learning can be measured, certified, and falsified.