Kernel Ordinary Differential Equations

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 as well. We demonstrate the efficacy of our method through numerous ODE examples.

Keywords:

• Component selection and smoothing operator
• High dimensionality
• Ordinary differential equations
• Smoothing spline analysis of variance
• Reproducing kernel Hilbert space

Supplementary Materials

The supplementary material contains proofs and additional numerical results for the main article.

Acknowledgments

We thank the Editor, the Associate Editor, and two referees for their constructive comments and suggestions.

Additional information

Funding

Xiaowu Dai’s research was partially supported by CDAR, Department of Economics, the University of California, Berkeley, and this work was done while Xiaowu Dai was visiting the Simons Institute for the Theory of Computing. Lexin Li’s research was partially supported by NSF grant DMS-1613137, and NIH grants R01AG061303, R01AG062542, and R01AG034570.