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.
[关键性质]
We characterize this ambiguity for planar drivers with bounded speed.
[结构刻画]
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.
[采样复杂度]
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.
[验证设置]
Synthetic experiments assess the certificate using admissible continuous completions and examine the role of path regularity.