Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics

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. Extensive experiments on simulated and real-world systems demonstrate that SKANODE achieves superior predictive accuracy, discovers physics-consistent dynamics, and reveals complex nonlinear behavior. Notably, it identifies hysteretic behavior in an F-16 aircraft and recovers a concise symbolic equation describing this phenomenon. SKANODE thus enables interpretable, data-driven discovery of physically grounded models for complex nonlinear dynamical systems.