Kernel Ordinary Differential Equations

Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this talk, I will discuss 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 construct confidence intervals for the estimated signal trajectories, and 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. Our proposal is also advantageous, in terms of statistical inference with noisy observations, compared to the existing physics-informed neural networks and sparsity-based methods.