# KAN-ODEs: Kolmogorov-Arnold Network Ordinary Differential Equations for Learning Dynamical Systems and Hidden Physics

**Authors:** Benjamin C. Koenig, Suyong Kim, Sili Deng  
**arXiv:** [2407.04192](https://arxiv.org/abs/2407.04192)  
**Date:** 2024-07-05  
**Summary page:** https://www.summarizepaper.com/en/paper/2407.04192v2/

## Key points

- Kolmogorov-Arnold networks (KANs) are a promising alternative to traditional multi-layer perceptrons (MLPs) in data-driven modeling.
- Incorporating KANs into the neural ordinary differential equation (ODE) framework, known as KAN-ODEs, offers advantages such as enhanced accuracy, faster neural scaling, improved interpretability and generalizability, and reduced parameter counts.
- A comprehensive study demonstrated the superior performance of KAN-ODEs over conventional Neural ODEs using the Lotka-Volterra predator-prey model as a test case.
- KAN-ODEs have shown the ability to learn symbolic source terms and generate complete solution profiles in complex and data-scarce scenarios, showcasing versatility in examples like wave propagation, shock formation, Schrödinger equation, and Allen-Cahn phase separation equation.
- The successful training of KAN-ODEs represents a significant breakthrough for leveraging this network architecture in scientific machine learning applications aimed at uncovering hidden physics phenomena.

## Summary

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In the realm of data-driven modeling, Kolmogorov-Arnold networks (KANs) have emerged as a promising alternative to traditional multi-layer perceptrons (MLPs). This innovative approach has been harnessed in a neural ordinary differential equation (ODE) framework, expanding its applicability to time-dependent and temporal grid-sensitive scenarios commonly encountered in dynamical systems and scientific machine learning applications. By incorporating KANs into the ODE framework, known as KAN-ODEs, researchers have unlocked a host of advantages over MLP-based models. These include enhanced accuracy, faster neural scaling, improved interpretability and generalizability, and reduced parameter counts. A comprehensive study was conducted to quantitatively demonstrate the superior performance of KAN-ODEs compared to conventional Neural ODEs using the classical Lotka-Volterra predator-prey model as a test case. The results not only validated the efficacy of KAN-ODEs but also highlighted their ability to learn symbolic source terms and generate complete solution profiles in complex and data-scarce scenarios. Examples such as wave propagation and shock formation, the intricate Schrödinger equation, and the Allen-Cahn phase separation equation showcased the versatility and robustness of the KAN-ODE framework. The successful training of KAN-ODEs signifies a significant breakthrough in leveraging this novel network architecture for scientific machine learning applications aimed at uncovering hidden physics and accurately predicting dynamic evolution. With contributions from Benjamin C. Koenig, Suyong Kim, and Sili Deng, this work represents a pivotal advancement in harnessing Kolmogorov-Arnold networks within an ODE framework for learning dynamical systems and unraveling hidden physics phenomena.