Understanding and modeling nonlinear dynamical systems is a fundamental challenge across science and engineering. Deep learning has shown remarkable potential for capturing complex system behavior, yet achieving models that are both accurate and physically interpretable remains difficult. To address this, we propose Structured Kolmogorov-Arnold Neural ODEs (SKANODEs), a framework that integrates structured state-space modeling with Kolmogorov-Arnold Networks (KANs). Within a Neural ODE architecture, SKANODE employs a fully trainable KAN as a universal function approximator to perform virtual sensing, recovering latent states that correspond to interpretable physical quantities such as displacements and velocities. Leveraging KAN's symbolic regression capability, SKANODE then extracts compact, interpretable expressions for the system's governing dynamics. Experiments on two canonical nonlinear oscillators and a real-world F-16 ground vibration dataset demonstrate that SKANODE reliably recovers physically meaningful latent displacement and velocity trajectories from acceleration measurements, identifies the correct governing nonlinearities--including the cubic stiffness in the Duffing oscillator and the nonlinear damping structure in the Van der Pol oscillator--and reveals hysteretic signatures in the F-16 interface dynamics through structured latent phase portraits and an interpretable symbolic model. Across all three cases, SKANODE provides more accurate and robust predictions than black-box NODE baselines and classical ARX and NARX identification, while producing equation-level descriptions of the learned nonlinear dynamics.

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