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

TL;DR

This paper tackles learning nonlinear dynamical systems from partial observations while preserving physical interpretability. It introduces Structured Kolmogorov-Arnold Neural ODEs (SKANODE), which combines a structured state-space with a trainable Kolmogorov-Arnold Network (KAN) to perform virtual sensing and end-to-end symbolic discovery of governing equations, with the latent states constrained to physically meaningful coordinates such as $x$ (displacement) and $v$ (velocity). A two-stage learning process uses $KAN_{approx}$ for latent dynamics and $KAN_{symbolic}$ for symbolic equation extraction, followed by calibration of the symbolic model within the Neural ODE. The authors establish an identifiability result under appropriate conditions and demonstrate superior predictive accuracy and interpretable dynamics on Duffing and Van der Pol oscillators, as well as hysteresis at a nonlinear interface in a real F-16 aircraft, highlighting the framework’s potential for physics-grounded discovery from indirect measurements. Overall, SKANODE bridges deep learning with interpretable, physics-consistent modeling, enabling reliable discovery and diagnostics in complex nonlinear systems.

Abstract

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.

Figures (6)

• Figure 1: A first-stage Kolmogorov--Arnold Network ($\text{KAN}_{\text{approx}}$) is employed as a universal function approximator within the proposed structured state-space Neural ODE framework to perform virtual sensing. Throughout training, the latent state variables and reconstructed observations are constrained to evolve as physically interpretable quantities---specifically displacement, velocity, and acceleration---enforced by the inductive biases encoded in the structured representation. Once coherent latent displacement and velocity states are learned, they are passed to a second network ($\text{KAN}_{\text{symbolic}}$), which performs symbolic equation discovery and extracts a closed-form expression for the governing dynamics. The resulting symbolic expression learned by ($\text{KAN}_{\text{symbolic}}$) is then substituted back into the Neural ODE, and the symbolic model is further trained to calibrate its coefficients, improving both the precision of the discovered governing equation and the predictive accuracy of system responses.
• Figure 2: Results on the Duffing oscillator. Top left: Symbolic governing equation discovered by $\text{KAN}_{\text{symbolic}}$, compared against the numerical baseline. The identified node receiving displacement input $x$ correctly captures the expected cubic nonlinearity. Top right: Predicted system observables (accelerations) obtained using the SKANODE framework. SKANODE accurately reconstructs accelerations directly from the inferred latent dynamics, without the need for a separate observation model. Bottom: Inferred latent state variables compared to ground truth. The latent states recovered by SKANODE correspond closely to physically meaningful displacement and velocity trajectories, unlike ANODE and SONODE, whose latent states remain abstract representations without clear physical interpretation.
• Figure 3: Results on the Van der Pol oscillator. Top left: Symbolic governing equation discovered by $\text{KAN}_{\text{symbolic}}$, compared against the numerical baseline. The identified node receiving displacement input $x$ correctly captures the expected quadratic nonlinearity. Top right: Predicted system observables (accelerations) obtained using the SKANODE framework. SKANODE accurately reconstructs accelerations directly from the inferred latent dynamics, without the need for a separate observation model. Bottom: Inferred latent state variables compared to ground truth. The latent states recovered by SKANODE correspond closely to physically meaningful displacement and velocity trajectories, unlike ANODE and SONODE, whose latent states remain abstract representations without clear physical interpretation.
• Figure 4: Overview of F16 aircraft and sensor location.
• Figure 5: Results on the F-16 aircraft. Top left: Symbolic governing equation discovered by $\text{KAN}_{\text{symbolic}}$. Top right: Predicted system observables (accelerations). Bottom: Inferred latent states visualized in phase portraits. The phase plot obtained by SKANODE exhibits distinct closed-loop patterns characteristic of hysteretic behavior, whereas the phase plot from ANODE yields abstract latent trajectories that lack clear physical interpretation.

Theorems & Definitions (3)

• Proposition 1: Identifiability of SKANODE
• Proposition 1: Identifiability of SKANODE
• proof