  We study nonparametric estimation in dynamical systems described by ordinary
differential equations (ODEs). Specifically, we focus on estimating the unknown
function $f \colon \mathbb{R}^d \to \mathbb{R}^d$ that governs the system
dynamics through the ODE $\dot{u}(t) = f(u(t))$, where observations $Y_{j,i} =
u_j(t_{j,i}) + \varepsilon_{j,i}$ of solutions $u_j$ of the ODE are made at
times $t_{j,i}$ with independent noise $\varepsilon_{j,i}$.
  We introduce two novel models -- the Stubble model and the Snake model -- to
mitigate the issue of observation location dependence on $f$, an inherent
difficulty in nonparametric estimation of ODE systems. In the Stubble model, we
observe many short solutions with initial conditions that adequately cover the
domain of interest. Here, we study an estimator based on multivariate local
polynomial regression and univariate polynomial interpolation. In the Snake
model we observe few long trajectories that traverse the domain on interest.
Here, we study an estimator that combines univariate local polynomial
estimation with multivariate polynomial interpolation.
  For both models, we establish error bounds of order $n^{-\frac{\beta}{2(\beta
+1)+d}}$ for $\beta$-smooth functions $f$ in an infinite dimensional function
class of H\"older-type and establish minimax optimality for the Stubble model
in general and for the Snake model under some conditions via comparison to
lower bounds from parallel work.
