  Distilling analytical models from data has the potential to advance our
understanding and prediction of nonlinear dynamics. Although discovery of
governing equations based on observed system states (e.g., trajectory time
series) has revealed success in a wide range of nonlinear dynamics, uncovering
the closed-form equations directly from raw videos still remains an open
challenge. To this end, we introduce a novel end-to-end unsupervised deep
learning framework to uncover the mathematical structure of equations that
governs the dynamics of moving objects in videos. Such an architecture consists
of (1) an encoder-decoder network that learns low-dimensional spatial/pixel
coordinates of the moving object, (2) a learnable Spatial-Physical
Transformation component that creates mapping between the extracted
spatial/pixel coordinates and the latent physical states of dynamics, and (3) a
numerical integrator-based sparse regression module that uncovers the
parsimonious closed-form governing equations of learned physical states and,
meanwhile, serves as a constraint to the autoencoder. The efficacy of the
proposed method is demonstrated by uncovering the governing equations of a
variety of nonlinear dynamical systems depicted by moving objects in videos.
The resulting computational framework enables discovery of parsimonious
interpretable model in a flexible and accessible sensing environment where only
videos are available.
