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.

翻译：从噪声数据中学习非参数常微分方程组 dot x = f(t,x) 是一个新兴的机器学习课题。我们利用成熟的再生核希尔伯特空间理论来定义函数f的候选集，使得该常微分方程的解存在且唯一。学习f的过程转化为在再生核希尔伯特空间中求解带约束的优化问题。我们提出了一种惩罚方法，通过迭代使用表示定理和欧拉近似来获得数值解。我们证明了关于x与其估计量之间L2距离的泛化界，并与最先进方法进行了实验对比。