Kernel Ordinary Differential Equations

TL;DR

This paper introduces a flexible kernel-based method for estimating and inferring ODEs from noisy data without assuming specific functional forms, applicable in both low- and high-dimensional settings.

Contribution

It develops a novel kernel ODE approach that handles unknown, nonlinear, and interactive functions, extending SS-ANOVA techniques with theoretical guarantees.

Findings

Estimation and selection are optimal in low- and high-dimensional regimes.

Method accurately recovers complex ODE systems from noisy observations.

Demonstrated effectiveness on various simulated ODE examples.

Abstract

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.