Neural ordinary differential equations (neural ODEs) have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of their optimal architecture remains elusive. In this work, we examine the interplay between their width $p$ and number of layer transitions $L$ (effectively the depth $L+1$). Specifically, we assess the model expressivity in terms of its capacity to interpolate either a finite dataset $D$ comprising $N$ pairs of points or two probability measures in $\mathbb{R}^d$ within a Wasserstein error margin $\varepsilon>0$. Our findings reveal a balancing trade-off between $p$ and $L$, with $L$ scaling as $O(1+N/p)$ for dataset interpolation, and $L=O\left(1+(p\varepsilon^d)^{-1}\right)$ for measure interpolation. In the autonomous case, where $L=0$, a separate study is required, which we undertake focusing on dataset interpolation. We address the relaxed problem of $\varepsilon$-approximate controllability and establish an error decay of $\varepsilon\sim O(\log(p)p^{-1/d})$. This decay rate is a consequence of applying a universal approximation theorem to a custom-built Lipschitz vector field that interpolates $D$. In the high-dimensional setting, we further demonstrate that $p=O(N)$ neurons are likely sufficient to achieve exact control.

翻译：神经常微分方程（neural ODEs）已从控制视角成为监督学习的自然工具，但其最优架构的完整理解仍不明确。本研究探讨了宽度$p$与层间过渡次数$L$（即有效深度$L+1$）之间的协同作用。具体而言，我们通过模型对包含$N个样本对的有限数据集$D$或$\mathbb{R}^d$中两个概率测度在Wasserstein误差容限$\varepsilon>0$下的插值能力来评估其表达能力。研究结果表明，$p$与$L$存在权衡关系：数据集插值时$L$按$O(1+N/p)$标度，测度插值时$L$按$O\left(1+(p\varepsilon^d)^{-1}\right)$标度。在自主情形（$L=0$）下需单独研究，本研究聚焦于数据集插值问题。我们针对$\varepsilon$-近似可控性的松弛问题进行分析，并建立了误差衰减率$\varepsilon\sim O(\log(p)p^{-1/d})$。该衰减率源于将通用近似定理应用于自定义的插值$D$的Lipschitz向量场。在高维设定中，我们进一步证明$p=O(N)$个神经元足以实现精确控制。