Abstract
<jats:p>Physics-informed neural networks (PINNs) face a fundamental parameter identifiability challenge when solving inverse problems governed by second-order ordinary differential equations under sparse displacement-only observations: different parameter values can produce nearly indistinguishable displacement responses, rendering the inverse problem ill-posed. This paper proposes derivative-observation-augmented PINNs (D-PINN), which explicitly incorporate first-order derivative observations into the PINN loss function to break this parameter degeneracy. We provide a theoretical analysis from three perspectives: a Sobolev-type error bound showing that derivative supervision controls displacement error, a Fisher information analysis proving that derivative observations add nonnegative information and improve local identifiability, and a residual-based stability estimate bounding parameter error by the physical residual and its derivatives. Numerical experiments on linear underdamped, forced near-resonance, and nonlinear Duffing oscillators with 10 random seeds and 20,000 training epochs demonstrate that D-PINN without a parameter prior reduces the damping coefficient relative error from 40% (standard PINN) to 11.7%. With a weak, non-oracle prior (μ0=3.2 vs. true value 4.0), the relative error further decreases to 2.1%, representing a 19-fold improvement over standard PINN. We also characterize sensitivity to prior quality, derivative observation source, and noise level, and discuss scenarios where D-PINN does not improve displacement fitting accuracy.</jats:p>