Axioms 2026, 15(8), 602; https://doi.org/10.3390/axioms15080602 (registering DOI) - 9 Aug 2026
Abstract
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.
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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 (
(This article belongs to the Special Issue Application of Machine Learning and Optimization Methods in Engineering Mathematics, 2nd Edition)
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