A Geometric Lens on Physics-Aligned Data Compression

In AI for Science, physics-informed losses are becoming popular to train learned compressors, but their rate-distortion consequences are poorly understood. We formalise this problem via a geometric framework, showing that physics-aware compression is governed by the interaction of two Riemannian structures in latent space: a Hessian-based physics sensitivity geometry induced by the physical observable, and a rate geometry induced by the entropy model. This theoretical view yields an explicit mechanism for error allocation: the codec concentrates precision along spectrally stiff and rate-expensive directions, while pushing uncertainty into directions that are weakly sensed by the physical observable. We prove fundamental limits from this alignment: (i) rate-efficient preservation is theoretically possible only when physical sensitivity is strongly anisotropic, and (ii) when physics and fidelity are not spectrally aligned, improving physical observables at fixed rate provably worsens standard distortion. We validate these predictions across chaotic fluid dynamics simulations, and introduce simple geometric alignment diagnostics that anticipate when physics-aligned training will succeed.