实验设置决定线性系统可识别性上限，部分动态仍可唯一还原
Limits of Learning Linear Dynamics from Experiments
- 基于实验初始状态和控制输入的几何分析，揭示可恢复信息的极限
- 即使系统整体不可识别，可观测子空间内的动态仍能唯一确定
- 为数据驱动建模提供理论边界，适合关注模型可靠性研究者
从数据中学习支配性动态是科学领域的常见目标，但仅在底层机制可识别时才有意义。实践中许多数据驱动方法隐含假设可识别性；当该假设不成立时，估计模型可能产生虚假预测和无效机理结论。经典可控线性时不变（LTI）系统的可识别性保证提供了充分条件——可控性和持续激励，但未说明这些条件不满足时可识别性是否仍成立，以及系统中哪些部分仍可识别。本文表明，实验设置（即实际的初始状态和控制输入）决定了从观测轨迹中可恢复信息的根本限制。我们发展了该限制的几何表征，并推导出与实验设置一致的所有系统闭式描述。关键证明显示，即使整个系统不可识别，实验可达子空间上的受限动态仍保持唯一确定。
原文摘要 · Abstract (English)
Learning governing dynamics from data is a common goal across the sciences, yet it is only well-posed when the underlying mechanisms are identifiable. In practice, many data-driven methods implicitly assume identifiability; when this assumption fails, estimated models can yield spurious predictions and invalid mechanistic conclusions. Classical identifiability guarantees for controlled linear time-invariant (LTI) systems provide sufficient conditions -- controllability and persistent excitation -- but leave open whether identifiability holds when these conditions fail, and which parts of the system remain identifiable without full identifiability. We show that the experimental setup, i.e., the realized initial state and control input, dictates a fundamental limit on the information recoverable from the observed trajectory. We develop a geometric characterization of this limit and derive a closed-form description of all systems consistent with the experimental setup. Crucially, we prove that even when the full system is not identifiable, the restricted dynamics on the subspace reachable by the experiment remain uniquely determined.
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