Structural Uncertainty in Physics-Informed Inverse Problems: The Hidden Risk in Scientific AI

Machine learning is increasingly used to infer physical laws from data, yet a model that reproduces observations accurately can still encode a physically non-identifiable coefficient function. This danger is particularly acute in physics-informed machine learning (PIML), where coefficient functions of governing equations—such as Hamiltonians or Lagrangians—are learned by minimizing physics-based predictive losses. We show that this failure can originate in the governing equations themselves: multiple, physically distinct coefficient functions can satisfy the same differential equations on the same observations, a situation that additional data cannot resolve when the non-uniqueness is structural. We introduce a rank-based criterion that diagnoses this structural uncertainty before training and evaluates whether candidate physical constraints restore identifiability. Specifically, we combine the numerical null space of the equation matrix $\mathbf{M}$ with the DNN tangent space to select the strength of a physical constraint without ground truth, based on the reduction of structural uncertainty. Experiments on Hamiltonian, Lagrangian, diffusion, and wave-kinetic systems demonstrate that predictive accuracy, structural identifiability, and physical-model recovery are distinct quantities.