The Dynamics of Discoverability: How Trajectories and Priors Shape Equation Recovery

How the dynamical regime of the observed system affects equation discovery has mainly been investigated through comparisons across systems. However, such comparisons vary both the equations and the dynamics, confounding the effect of the regime with the difficulty of recovering the equations symbolically. We separate the two by varying the forcing of Lorenz-84, moving its fully observed post-transient trajectories through fixed-point, periodic, and chaotic regimes while preserving the equations' functional form. Within each regime, we separately vary the amount of data, the noise, and prior knowledge of which terms the equations contain. We then measure how well two complementary approaches, sparse regression over a fixed library of candidate functions (SINDy) and an evolutionary search over symbolic expression trees (PySR), recover the true equations' terms and coefficients. We find that recovery depends on whether the sampled states distinguish combinations of candidate functions: equations remain poorly recovered from fixed-point data even when the candidate set contains only the true terms. We link the effect of the dynamics on both algorithms to one object: the moment matrix of the candidate functions under the invariant measure of the regime. Small eigenvalues mark weakly distinguishable combinations of candidate functions: we show that more data, less noise, and more prior knowledge can mitigate the resulting recovery difficulties, while a zero eigenvalue makes distinct equations indistinguishable on the visited states. Hence, a more precise prior needs less informative data, with consequences for data collection, method design, and evaluation.