Learning nonparametric systems of Ordinary Differential Equations (ODEs) $\dot x = f(t,x)$ from noisy and sparse 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 $L^2$ distance between $x$ and its estimator. Experiments are provided for the FitzHugh Nagumo oscillator and for the prediction of the Amyloid level in the cortex of aging subjects. In both cases, we show competitive results when compared with the state of the art.

翻译：普通差异平方(ODEs)$\dot x = f(t,x)美元 = f(t,x) 来自噪音和稀少的数据的学习非参数系统是一个新兴的机器学习主题,我们使用开发完善的复制Kernel Hilbert空间理论(RKHS)来确定存在和独特的ODE解决方案的美元候选人。学习美元包括解决在RKHS中限制优化的问题。我们建议一种惩罚方法,反复使用代表理论和Euler近似法来提供数字解决方案。我们证明,对于美元与其估计仪之间的距离,我们是一个通用化值。为FitzHugh Nagumo振动器提供了实验,并用于预测老化主体皮层的埃米洛德水平。在这两种情况下,我们都显示与艺术状态相比的竞争性结果。