Recovering Governing Equations from Solution Data: Identifiability Bounds for Linear and Nonlinear ODEs

Scientific machine learning often aims to recover governing equations from observed data. However, a rigorous theoretical foundation for when a ground-truth ODE can be uniquely and stably identified from solution trajectories remains missing. This paper addresses this gap by formalizing the identification problem and providing quantitative bounds on the sample complexity required to recover the governing dynamics.

The authors introduce the Hausdorff distance between solution sets as the natural metric for comparing differential equations. This metric captures the worst-case separation between trajectories over admissible initial conditions, effectively encoding the minimax structure of the identification task. By utilizing this metric, the authors analyze the relationship between the distance of the underlying structure equations (e.g., the vector field $f$ in $\dot{x} = f(x)$) and the Hausdorff distance of the resulting solution behaviors. They derive metric entropy estimates and sample complexity bounds for several classes of ODEs, including linear, Lipschitz, Hölder-continuous, and polynomial systems.

The study demonstrates that the Hausdorff distance provides a stable and meaningful way to compare ODEs. For linear ODEs, the authors show that the Hausdorff distance is directly proportional to the norm of the difference between the system matrices, scaled by factors related to the time horizon and the magnitude of the initial conditions. For Lipschitz and Hölder-continuous classes, they establish upper and lower bounds that characterize how the structural differences in the vector fields manifest in the solution space. These results provide a theoretical guarantee for the uniqueness and stability of equation discovery algorithms.