Nonparametric Estimation of Ordinary Differential Equations: Snake and Stubble

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.