论文标题
内核普通微分方程
Kernel Ordinary Differential Equations
论文作者
论文摘要
普通的微分方程（ODE）广泛用于建模科学中的生物学和物理过程。在本文中，我们提出了一种新的基于内核的方法，以估算和推断ODE的噪声观察。我们不认为ode中的功能形式已知，也不会将其限制为线性或添加剂，因此我们允许成对相互作用。我们执行稀疏的估计以选择个体功能，并为估计的信号轨迹构建置信区间。我们在低维和高维度设置下建立了内核ode的估计最优性和选择一致性，在该设置中，未知功能的数量可能比样本量更小或更大。我们的建议基于对方差的平滑样条分析（SS-ANOVA）框架的基础，但是解决了尚未完全解决的几个重要问题，因此也扩大了现有的SS-Anova的范围。我们通过众多ode示例证明了我们方法的功效。
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.
