From Observation Prediction to Control Coordinates: Learning Differentially Flat Latent Representations

Latent dynamics models enable control of systems from high-dimensional observations. Existing methods primarily emphasize predictive accuracy, while control depends on the relationships among the learned representation, the underlying physical state, and the control input. Accurate observation-level prediction alone may be insufficient for effective control, particularly when observations do not uniquely determine the physical state. We address this gap by introducing differential flatness as a structural bias. For differentially flat systems, we establish a sufficient observation condition: observations need only distinguish states with distinct flat outputs, whose finite-order derivatives enable state and input recovery. The model combines a flat-jet encoder and structured latent dynamics, trained jointly without supervision on the true flat coordinates. This representation supports direct trajectory planning in learned flat coordinates and translation into control inputs. Experiments on an analytically tractable flat system and image-based differentially flat systems under incomplete observations show that our model learns coordinates with differential flatness properties, improves prediction and control accuracy over prediction-centric baselines, particularly with limited data, and reduces computation for closed-loop control.