Ordinary Differential Equations (ODEs) have recently gained a lot of attention in machine learning. However, the theoretical aspects, e.g., identifiability and asymptotic properties of statistical estimation are still obscure. This paper derives a sufficient condition for the identifiability of homogeneous linear ODE systems from a sequence of equally-spaced error-free observations sampled from a single trajectory. When observations are disturbed by measurement noise, we prove that under mild conditions, the parameter estimator based on the Nonlinear Least Squares (NLS) method is consistent and asymptotic normal with $n^{-1/2}$ convergence rate. Based on the asymptotic normality property, we construct confidence sets for the unknown system parameters and propose a new method to infer the causal structure of the ODE system, i.e., inferring whether there is a causal link between system variables. Furthermore, we extend the results to degraded observations, including aggregated and time-scaled ones. To the best of our knowledge, our work is the first systematic study of the identifiability and asymptotic properties in learning linear ODE systems. We also construct simulations with various system dimensions to illustrate the established theoretical results.

翻译：常微分方程（ODEs）近来在机器学习领域受到广泛关注。然而，其理论方面，例如统计估计的可辨识性与渐近性质，仍不明确。本文推导了从单条轨迹等间距无误差观测序列中辨识齐次线性常微分方程系统的充分条件。当观测受到测量噪声干扰时，我们证明在温和条件下，基于非线性最小二乘法（NLS）的参数估计量具有一致性，且呈渐近正态分布，收敛速度为$n^{-1/2}$。基于该渐近正态性，我们为未知系统参数构建了置信集，并提出了一种推断常微分方程系统因果结构的新方法，即推断系统变量间是否存在因果关联。此外，我们将结果推广至退化观测情形，包括聚合观测与时间缩放观测。据我们所知，本研究首次系统性地探讨了学习线性常微分方程系统的可辨识性与渐近性质。我们还通过不同系统维度的仿真实验验证了所建立的理论结果。