Same Dynamics, Different Structures in Energy-Preserving Operator Fits

Energy-preserving reduced-order models can reproduce observed trajectories even when the coefficients describing their internal structure are not uniquely determined. We study this structural identifiability problem for quadratic energy-preserving dynamics. For the model class considered here, a single quadratic energy leaves a coefficient null space of dimension $\binom{d}{3}$, independent of the amount of trajectory data. On published 2D Burgers data, two admissible fits differ by 68% in coefficient norm while their vector fields differ by only $6.2 \cdot 10^{-14}$. For Lie–Poisson models, we further determine whether these undetectable coefficient changes are only changes of basis or can produce non-isomorphic Lie brackets with the same dynamics. This leads to an identifiability certificate that determines whether enforcing the Jacobi identity can recover the coefficients or whether a second experiment with a different energy is required. Under Gaussian observation noise, we calibrate the numerical rank test using the predicted coefficient uncertainty and abstain when the rank is unresolved. A high-probability bound then gives the trajectory count required for a specified coefficient error.