提出新度量方法，定量分析从数据中识别微分方程所需的最少观测数。
Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs
- 用解集的豪斯多夫距离衡量方程差异，体现最坏情况下的区分能力。
- 给出线性与非线性常微分方程的可识别性边界，明确需多少数据才能可靠恢复。
- 适用于科学机器学习中方程发现任务，尤其关注样本复杂度问题。
从观测解数据中学习控制方程是科学机器学习的核心挑战，但目前对从多个解观测中唯一且稳定识别真实常微分方程（ODE）的理论条件仍缺乏系统研究，且尚未有文献提供此类学习任务的量化样本复杂度分析。为填补该空白，本文引入解集上的豪斯多夫距离作为比较微分方程的自然度量，因其能捕捉在所有可接受初值条件下两方程间的最大分离程度，从而体现识别问题的极小极大结构。我们针对广泛类别的结构化方程——从线性ODE到具有Lipschitz（Hölder）连续向量场的非线性类——建立了可识别性边界，精确刻画了两个不同方程在解数据下可被区分的条件。基于该度量，我们推导了相关ODE类的度量熵估计，并分析了样本复杂度界，量化了可靠恢复控制方程所需解观测的数量。
原文摘要 · Abstract (English)
Learning governing equations from observed solution data is a fundamental challenge in scientific machine learning, yet the theoretical conditions under which a ground-truth ODE can be uniquely and stably identified from multiple solution observations remain largely undeveloped, and no quantitative analysis of the sample complexity of such learning tasks exists in the literature. To address this gap, we introduce the Hausdorff distance on solution sets as the natural metric for comparing differential equations, since it captures the worst-case separation between two equations over all admissible initial conditions and thus encodes the minimax structure of the identification problem. We establish identifiability bounds for governing ODEs across a wide class of structure equations--ranging from linear ODEs to nonlinear classes with Lipschitz (Hölder)-continuous vector fields--characterizing precisely when two distinct equations can be distinguished from solution data. Using this metric, we derive metric entropy estimates for the relevant ODE classes and analyze sample complexity bounds, quantifying how many solution observations are needed to reliably recover the governing equation.
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