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.

神经常微分方程（neural ODE）的宽度 p 和层转换的数量 L 之间存在平衡折衷，通过对数据集插值和度量插值进行评估，发现对于数据集插值，L 随着 O（1 + N /p）的规模变化，对于度量插值，L 为 O（1 +（pε ^ d）^ -1）。在自主情况下，L=0 需要单独研究，关注数据集插值问题。我们解决了 ε- 近似可控性的放宽问题，并建立 ε 的误差减小率为 O（log（p）p ^ -1 /d）。在高维度设置中，我们进一步证明 p = O（N）个神经元可能足以实现精确控制。