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

arXiv: https://arxiv.org/abs/2506.18339 · HTML: https://mlsanity.com/paper/2506.18339/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 65 references that are in the map.
- First posted on arXiv 23 Jun 2025.

## Builds on (most cited first, 30 of 40)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Fundamentals of Recurrent Neural Network (RNN) and Long Short-Term Memory (LSTM) Network](https://mlsanity.com/paper/1808.03314/) (2018)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [KalmanNet: Neural Network Aided Kalman Filtering for Partially Known Dynamics](https://mlsanity.com/paper/2107.10043/) (2021)
- [Dynamical Variational Autoencoders: A Comprehensive Review](https://mlsanity.com/paper/2008.12595/) (2020)
- [Neural SDE: Stabilizing Neural ODE Networks with Stochastic Noise](https://mlsanity.com/paper/1906.02355/) (2019)
- [Integration of Neural Network-Based Symbolic Regression in Deep Learning for Scientific Discovery](https://mlsanity.com/paper/1912.04825/) (2019)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Deep Learning via Dynamical Systems: An Approximation Perspective](https://mlsanity.com/paper/1912.10382/) (2019)
- [On Second Order Behaviour in Augmented Neural ODEs](https://mlsanity.com/paper/2006.07220/) (2020)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [Physics-Integrated Variational Autoencoders for Robust and Interpretable Generative Modeling](https://mlsanity.com/paper/2102.13156/) (2021)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [Integrating Expert ODEs into Neural ODEs: Pharmacology and Disease Progression](https://mlsanity.com/paper/2106.02875/) (2021)
- [Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics](https://mlsanity.com/paper/2110.10249/) (2021)
- [Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems](https://mlsanity.com/paper/2010.14685/) (2020)
- [Heavy Ball Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.04840/) (2021)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [Deep Generative Symbolic Regression with Monte-Carlo-Tree-Search](https://mlsanity.com/paper/2302.11223/) (2023)
- [Variational Integrator Networks for Physically Structured Embeddings](https://mlsanity.com/paper/1910.09349/) (2019)
- [StableSSM: Alleviating the Curse of Memory in State-space Models through Stable Reparameterization](https://mlsanity.com/paper/2311.14495/) (2023)
- [Learning Interpretable Deep State Space Model for Probabilistic Time Series Forecasting](https://mlsanity.com/paper/2102.00397/) (2021)
- [ChemNODE: A Neural Ordinary Differential Equations Approach for Chemical Kinetics Solvers](https://mlsanity.com/paper/2101.04749/) (2020)
- [Predicting Ordinary Differential Equations with Transformers](https://mlsanity.com/paper/2307.12617/) (2023)

## Built on by (most cited first, 4 of 4)

- [Empirical Stability Analysis of Kolmogorov-Arnold Networks in Hard-Constrained Recurrent Physics-Informed Discovery](https://mlsanity.com/paper/2602.09988/) (2026)
- [Holomorphic Neural ODEs with Kolmogorov-Arnold Networks for Interpretable Discovery of Complex Dynamics](https://mlsanity.com/paper/2605.22235/) (2026)
- [An Embedded RISC-V Evaluation of Kolmogorov--Arnold Networks in Hard-Constrained Recurrent Physics-Informed Models](https://mlsanity.com/paper/2608.00737/) (2026)
- [FluxDisco: Symbolic Regression for Stoichiometric Dynamical Systems via Monte Carlo Graph Search](https://mlsanity.com/paper/2609.05207/) (2026)

## Abstract (arXiv)

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.