  Ordinary differential equation (ODE) is widely used in modeling biological
and physical processes in science. In this article, we propose a new
reproducing kernel-based approach for estimation and inference of ODE given
noisy observations. We do not assume the functional forms in ODE to be known,
or restrict them to be linear or additive, and we allow pairwise interactions.
We perform sparse estimation to select individual functionals, and construct
confidence intervals for the estimated signal trajectories. We establish the
estimation optimality and selection consistency of kernel ODE under both the
low-dimensional and high-dimensional settings, where the number of unknown
functionals can be smaller or larger than the sample size. Our proposal builds
upon the smoothing spline analysis of variance (SS-ANOVA) framework, but
tackles several important problems that are not yet fully addressed, and thus
extends the scope of existing SS-ANOVA too. We demonstrate the efficacy of our
method through numerous ODE examples.
