Limits of Learning Linear Dynamics from Experiments

2026-05-12

AI Summary

This work addresses the fundamental limitations imposed by unidentifiability in learning linear dynamical systems from experimental data, which can lead to erroneous predictions and misinterpretations of underlying mechanisms. The study investigates how initial states and control inputs constrain the information recoverable from linear time-invariant systems and introduces a geometric characterization of identifiability: even when the full system is unidentifiable, the dynamics restricted to the experimentally reachable subspace are uniquely determined. By integrating controllability analysis, persistent excitation conditions, and geometric reasoning, the authors derive closed-form expressions for all systems consistent with observed trajectories, precisely delineating the identifiable component under given experimental conditions. This theoretical framework provides rigorous guarantees for data-driven modeling and safeguards against flawed inferences arising from inherent unidentifiability.

Abstract

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.

Problem

• identifiability
• linear dynamics
• experimental setup
• LTI systems
• reachable subspace

Innovation

• identifiability
• linear time-invariant systems
• reachable subspace
• geometric characterization
• data-driven dynamics