Learning nonparametric ordinary differential equations from noisy data

TL;DR

This paper introduces a nonparametric approach to learning ODE systems from noisy data using RKHS theory, proposing a penalty method with theoretical guarantees and experimental validation.

Contribution

It develops a novel RKHS-based framework for nonparametric ODE learning, including a penalty method and generalization bounds, advancing the state-of-the-art in noisy data scenarios.

Findings

The proposed method accurately learns ODEs from noisy data.

Theoretical generalization bounds are established.

Experimental results outperform existing approaches.

Abstract

Learning nonparametric systems of Ordinary Differential Equations (ODEs) dot x = f(t,x) from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for f for which the solution of the ODE exists and is unique. Learning f consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the L2 distance between x and its estimator and provide experimental comparisons with the state-of-the-art.