Ordinary differential equation (ODE) is an important tool to study the dynamics of a system of biological and physical processes. A central question in ODE modeling is to infer the significance of individual regulatory effect of one signal variable on another. However, building confidence band for ODE with unknown regulatory relations is challenging, and it remains largely an open question. In this article, we construct post-regularization confidence band for individual regulatory function in ODE with unknown functionals and noisy data observations. Our proposal is the first of its kind, and is built on two novel ingredients. The first is a new localized kernel learning approach that combines reproducing kernel learning with local Taylor approximation, and the second is a new de-biasing method that tackles infinite-dimensional functionals and additional measurement errors. We show that the constructed confidence band has the desired asymptotic coverage probability, and the recovered regulatory network approaches the truth with probability tending to one. We establish the theoretical properties when the number of variables in the system can be either smaller or larger than the number of sampling time points, and we study the regime-switching phenomenon. We demonstrate the efficacy of the proposed method through both simulations and illustrations with two data applications.

翻译：常微分方程是研究生物和物理过程系统动力学的重要工具。在常微分方程建模中，一个核心问题是推断一个信号变量对另一个变量的个体调控效应的显著性。然而，在未知调控关系下为常微分方程构建置信带极具挑战性，这仍然是一个很大程度上悬而未决的问题。本文针对含有未知泛函和含噪数据观测的常微分方程，构建了个体调控函数的正则化后置信带。我们的方法尚属首创，其构建基于两个新颖要素：第一是一种新的局部核学习法，它将再生核学习与局部泰勒近似相结合；第二是一种新的去偏方法，用于处理无穷维泛函和额外的测量误差。我们证明所构建的置信带具有理想的渐近覆盖概率，且恢复出的调控网络以趋近于1的概率逼近真实网络。当系统中的变量数小于或大于采样时间点数时，我们均建立了理论性质，并研究了状态切换现象。通过仿真实验和两个数据应用实例，我们展示了所提出方法的有效性。