  Ordinary differential equation (ODE) is an important tool to study the
dynamics of a system of biological and physical processes. A central question
in ODE modeling is to infer the significance of individual regulatory effect of
one signal variable on another. However, building confidence band for ODE with
unknown regulatory relations is challenging, and it remains largely an open
question. In this article, we construct post-regularization confidence band for
individual regulatory function in ODE with unknown functionals and noisy data
observations. Our proposal is the first of its kind, and is built on two novel
ingredients. The first is a new localized kernel learning approach that
combines reproducing kernel learning with local Taylor approximation, and the
second is a new de-biasing method that tackles infinite-dimensional functionals
and additional measurement errors. We show that the constructed confidence band
has the desired asymptotic coverage probability, and the recovered regulatory
network approaches the truth with probability tending to one. We establish the
theoretical properties when the number of variables in the system can be either
smaller or larger than the number of sampling time points, and we study the
regime-switching phenomenon. We demonstrate the efficacy of the proposed method
through both simulations and illustrations with two data applications.
