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

Abstract: Understanding and modeling nonlinear dynamical systems is a fundamental problem across scientific and engineering domains. While deep learning has demonstrated remarkable potential for learning complex system behavior, achieving models that are both highly accurate and physically interpretable remains a major challenge. To address this, we propose Structured Kolmogorov-Arnold Neural ODEs (SKANODEs), a novel framework that integrates structured state-space modeling with the Kolmogorov-Arnold Network (KAN). SKANODE first employs a fully trainable KAN as a universal function approximator within a structured Neural ODE framework to perform virtual sensing, recovering latent states that correspond to physically interpretable quantities such as positions and velocities. Once this structured latent representation is established, we exploit the symbolic regression capability of KAN to extract compact and interpretable expressions for the system's governing dynamics. The resulting symbolic expression is then substituted back into the Neural ODE framework and further calibrated through continued training to refine its coefficients, enhancing both the precision of the discovered equations and the predictive accuracy of system responses. Extensive experiments on both simulated and real-world systems demonstrate that SKANODE achieves superior performance while offering interpretable, physics-consistent models that uncover the underlying mechanisms of nonlinear dynamical systems.