Interplay between depth and width for interpolation in neural ODEs

Neural ordinary differential equations have emerged as a natural tool for supervised learning from a control perspective, yet a complete understanding of the role played by their architecture remains elusive. In this work, we examine the interplay between the width $p$ and the number of transitions between layers $L$ (corresponding to a depth of $L+1$). Specifically, we construct explicit controls interpolating either a finite dataset $D$, comprising $N$ pairs of points in $R^d$, or two probability measures within a Wasserstein error margin $\varepsilon >0$. Our findings reveal a balancing trade-off between $p$ and $L$, with $L$ scaling as $1+O(N/p)$ for data interpolation, and as $1+O\left(p^{-1}+{(1+p)}^{-1}{\varepsilon }^{-d}\right)$ for measures.

In the high-dimensional and wide setting where $d,p>N$, our result can be refined to achieve $L=0$. This naturally raises the problem of data interpolation in the autonomous regime, characterized by $L=0$. We adopt two alternative approaches: either controlling in a probabilistic sense, or by relaxing the target condition. In the first case, when $p=N$ we develop an inductive control strategy based on a separability assumption whose probability increases with $d$. In the second one, we establish an explicit error decay rate with respect to $p$ which results from applying a universal approximation theorem to a custom-built Lipschitz vector field interpolating $D$.