On Vector Field Reconstruction from Noisy ODE in High Ambient Dimension

This work investigates the nonparametric estimation of the vector field of a noisy Ordinary Differential Equation (ODE) in high-dimensional ambient spaces, under the assumption that the initial conditions are sampled from a lower-dimensional structure. Specifically, let \( f:\mathbb{R}^{D}\to\mathbb{R}^{D} \) denote the vector field of the autonomous ODE \( y' = f(y) \). We observe noisy trajectories \( \tilde{y}_{X_i}(t_j) = y_{X_i}(t_j) + \varepsilon_{i,j} \), where \( y_{X_i}(t_j) \) is the solution at time \( t_j \) with initial condition \( y(0)=X_i \), the \( X_i \) are drawn from a \((a,b)\)-standard distribution \( \mu \), and \( \varepsilon_{i,j} \) denotes noise. From a minimax perspective, we study the reconstruction of \( f \) within the envelope of trajectories generated by the support of \( \mu \). We proposed an estimator combining flow reconstruction with derivative estimation techniques from nonparametric regression. Under mild regularity assumptions on \( f \), we establish convergence rates that depend on the temporal resolution, the number of initial conditions, and the parameter \( b \), which controls the mass concentration of \( \mu \). These rates are then shown to be minimax optimal (up to logarithmic factors) and illustrate how the proposed approach mitigates the curse of dimensionality. Additionally, we illustrate the computational and statistical efficiency of our estimator through numerical experiments.