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

2025-06-23

AI Summary

Addressing the longstanding trade-off between modeling accuracy and physical interpretability in nonlinear dynamical systems, this paper proposes the Structured Kolmogorov–Arnold Neural Ordinary Differential Equation (KA-NeuODE) framework. The method integrates trainable Kolmogorov–Arnold Networks (KANs) with structurally constrained neural ODEs, enabling physically consistent dynamics modeling via latent-state reconstruction, and couples differentiable symbolic regression for end-to-end joint optimization—simultaneously learning high-fidelity state evolution and discovering concise, interpretable symbolic dynamical equations. Its key innovation lies in unifying KANs’ strong nonlinear approximation capability, ODEs’ physical priors, and symbolic regression’s interpretability within a fully differentiable training paradigm, thereby supporting virtual sensing and closed-loop optimization of equation coefficients. Evaluated on multiple benchmark dynamical systems, KA-NeuODE achieves over 40% reduction in prediction error and significantly improves symbolic equation discovery accuracy, yielding dynamical models that are both physically meaningful and high-performing.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Search and Optimization: Learning to Search

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

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.

Problem

• Develop interpretable models for nonlinear dynamical systems
• Combine structured state-space modeling with Kolmogorov-Arnold Networks
• Extract symbolic expressions for system dynamics from data

Innovation

• Structured Neural ODEs with Kolmogorov-Arnold Networks
• Symbolic regression for interpretable dynamics discovery
• Virtual sensing to recover physically interpretable states