Limits of Learning Linear Dynamics from Experiments

TL;DR

This paper investigates the fundamental limits of learning linear system dynamics from experiments, showing that experimental conditions determine what parts of the system can be identified, even when full identifiability fails.

Contribution

It provides a geometric framework to characterize what system components are identifiable based on experimental setup, extending classical identifiability results.

Findings

Experimental setup dictates the information recoverable from data.

Even when full system is not identifiable, reachable subspace dynamics are uniquely determined.

A closed-form description of all systems consistent with the experimental conditions is derived.

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.