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：从含噪数据中学习非参数常微分方程组（ODEs）\(\dot{x} = f(t,x)\) 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（RKHS）理论来定义候选函数 \(f\)，确保该常微分方程解的存在性与唯一性。学习 \(f\) 的过程可转化为在RKHS中求解一个约束优化问题。我们提出了一种惩罚方法，该方法通过迭代使用表示定理与欧拉近似来提供数值解。我们证明了估计量 \(\hat{x}\) 与真实解 \(x\) 之间 \(L^2\) 距离的泛化界，并与当前最先进方法进行了实验对比。