Nonparametric Estimation of Ordinary Differential Equations: Snake and Stubble
Christof Schötz
TL;DR
This work tackles nonparametric drift estimation in autonomous ODEs from noisy observations by introducing two restricted data-collection schemes, the Stubble and Snake models, that circumvent the locality problem where information about $f$ is concentrated near the unknown trajectory $u(t)$. Each model yields estimators that combine nonparametric regression (local polynomial type) with interpolation steps (univariate/poly- interpolation in Stubble, multivariate polynomial interpolation in Snake) to recover the drift function $f^*$. The authors prove minimax-optimal convergence rates for these estimators on Hölder-type function classes, with precise bounds of the form $n^{- rac{eta}{2(eta+1)+d}}$ (and refined variants depending on the model and regime), and they connect these rates to standard nonparametric regression benchmarks via black-box risk bounds. The results establish the theoretical optimality of the Stubble estimator generally and provide minimax-optimality for the Snake estimator under certain sampling conditions, highlighting the complementary nature of the two models. Collectively, the work advances nonparametric ODE learning by offering rigorous, rate-optimal strategies that respect the trajectory-driven information structure and by clarifying how trajectory coverage and sampling design govern estimation limits and practical performance.
Abstract
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 $n^{-\fracβ{2(β+1)+d}}$ for $β$-smooth functions $f$ in an infinite dimensional function class of Hölder-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.