# Kernel Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2008.02915 · HTML: https://mlsanity.com/paper/2008.02915/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Aug 2020.

## Built on by (most cited first, 5 of 5)

- [Scientific Inference With Interpretable Machine Learning: Analyzing Models to Learn About Real-World Phenomena](https://mlsanity.com/paper/2206.05487/) (2022)
- [Learning nonparametric ordinary differential equations from noisy data](https://mlsanity.com/paper/2206.15215/) (2022)
- [Post-Regularization Confidence Bands for Ordinary Differential Equations](https://mlsanity.com/paper/2110.12510/) (2021)
- [Sparse Equation Matching: A Derivative-Free Learning for General-Order Dynamical Systems](https://mlsanity.com/paper/2507.20072/) (2025)
- [Robust Learning of Heterogeneous Dynamic Systems](https://mlsanity.com/paper/2604.05285/) (2026)

## Abstract (arXiv)

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.