We study supervised regression with neural ODEs (NODEs) from a control-theoretic perspective to derive explicit population-risk bounds. We focus on a widely used class of non-autonomous models with constant parameters and explicit time dependence, which we call semi-autonomous NODEs (SA-NODEs). We constructively prove that SA-NODEs are capable of \emph{exact} interpolation of admissible finite datasets, and even satisfy a stronger property that we call \emph{simultaneous cell controllability} (SCC): their flows can map prescribed disjoint cells into arbitrarily small target balls. This property is the mechanism that upgrades interpolation into quantitative generalization, by allowing SA-NODEs to emulate piecewise-constant nonparametric estimators. Consequently, our risk bounds recover the rates of histogram and nearest-neighbor estimators, provided the network width satisfies a conservative scaling with the sample size. Numerical experiments show that trained SA-NODEs achieve competitive -- often lower -- test errors than these baselines. Finally, we show that the explicit time dependence is essential. Although two-layer autonomous NODEs can interpolate geometrically nondegenerate datasets, structural obstructions prevent them from achieving SCC. These limitations, further confirmed numerically, support the view that SA-NODEs provide a minimal effective architecture for learning.

翻译：我们从控制论视角研究神经ODE的监督回归问题，推导显式的总体风险界。重点关注一类广泛使用的非自治模型，这类模型具有恒定参数和显式时间依赖性，我们称之为半自治神经ODE（SA-NODE）。我们构造性地证明，SA-NODE能够对可行的有限数据集实现精确插值，且满足我们称为同时细胞可控性（SCC）的更强性质：其流可将指定的不相交细胞映射到任意小的目标球内。该性质通过允许SA-NODE模拟分段常数非参数估计器，将插值能力提升为定量泛化能力。据此，当网络宽度满足与样本量保守缩放时，我们的风险界能够恢复直方图估计器和最近邻估计器的收敛速率。数值实验表明，训练后的SA-NODE可实现具有竞争力（通常更低）的测试误差。最后，我们证明显式时间依赖性不可或缺：尽管两层自治神经ODE能够对几何非退化数据集进行插值，但结构障碍阻止其实现SCC。这些局限性经数值实验进一步证实，支持SA-NODE是学习所需的最简有效架构这一观点。