Physics from Video: Identifiability of Time-Invariant Second-Order ODEs under Minimal Trajectory Conditions
Provides theoretical guarantees for extracting interpretable physical constants from video without pixel reconstruction, addressing a core challenge in video-based world models.
The paper proves that second-order linear ODE parameters can be uniquely recovered from raw pixels under a level-set slope-coverage condition, with underdamped systems identifiable from a single video clip and other regimes requiring three diverse trajectories. A variance-floor regularizer stabilizes training and prevents latent collapse.
Bridging the gap between visual realism and physical understanding is a core challenge for video-based world models. We study the structural identifiability of continuous-time physical laws from raw pixels, focusing on whether an encoder-only pipeline can uniquely recover the parameters of second-order linear ODEs. We prove that a level-set slope-coverage condition ensures the learned latent space is locally affine to the true physical state, enabling exact parameter recovery. Our theory provides the first characterization of minimal data requirements across damping regimes, establishing that underdamped systems are identifiable from a single video clip, whereas other regimes require three diverse trajectories. We further introduce a variance-floor regularizer to stabilize the decoder-free objective and prevent latent collapse. Validated on synthetic and real-world data, our approach demonstrates that interpretable physical constants can be reliably estimated from video without the need for compute-intensive pixel reconstruction, ensuring both physical correctness and transparency. Code is available at https://github.com/wenjiewang3/PhysicsFromVideo.