A Sobolev–Information Perspective on Derivative-Observation-Augmented PINNs for Parameter Identification of Second-Order Dynamical Systems

Identifying parameters of dynamical systems from sparse measurements is a core task in structural health monitoring and vibration engineering. For second-order oscillators, standard physics-informed neural networks (PINNs) struggle because different parameter values can produce nearly identical displacement records, making the inverse problem ill-posed. We propose the derivative-observation-augmented PINN (D-PINN), which incorporates velocity measurements into the training loss to resolve this degeneracy. Three theoretical results support the method: a Sobolev-type inequality proves that constraining the velocity error automatically bounds the displacement error; a Fisher information analysis shows that velocity observations increase the information available for parameter estimation; and a residual-based estimate bounds the parameter error in terms of the solution accuracy and its derivatives. Experiments on linear, forced near-resonance, and Duffing oscillators (10 random seeds, 20,000 epochs) show that D-PINN reduces the damping coefficient relative error from 40% to 11.7% without any parameter prior. With a weak prior (μ0=3.2, a 20% deviation from the true value 4.0), the error drops further to 2.1%, a 19-fold improvement over standard PINN. We also analyze sensitivity to prior quality, derivative observation source, and measurement noise, and identify scenarios where derivative observations do not improve displacement fitting.