The Information Limits of Neural Controlled Differential Equations Under Sparse Observations

Neural controlled differential equations (Neural CDEs) evolve hidden states along continuous paths constructed by interpolating discrete observations. Sparse measurements leave the input between observation times undetermined, and admissible continuous paths consistent with the same data can produce different terminal states. We characterize this ambiguity for planar drivers with bounded speed. The set of compatible terminal states admits a leading-order zonotopic approximation determined by feasible signed areas and transported Lie brackets. Its projection onto a scalar prediction yields a computable, target-specific ambiguity certificate. We also study the number of measurements needed to reduce the remaining uncertainty. For nondegenerate planar CDEs, the optimal worst-case radius over deterministic adaptive sampling policies scales as $N^{-1}$ when speed is bounded and $N^{-2}$ when both speed and acceleration are bounded, where $N$ is the number of observation intervals. Synthetic experiments assess the certificate using admissible continuous completions and examine the role of path regularity.