  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.
