Kolmogorov–Arnold-Informed neural network

A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov–Arnold Networks

authored by: Yizheng Wang, Jia Sun, Jinshuai Bai, Cosmin Anitescu, Mohammad Sadegh Eshaghi, Xiaoying Zhuang, Timon Rabczuk, Yinghua Liu
Abstract: AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov–Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov–Arnold-Informed Neural Network (KINN) for solving forward and inverse problems. We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP regarding accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.
Organisation(s): Institute of Photonics
External Organisation(s): Tsinghua University
Bauhaus-Universität Weimar
CNPC Engineering Technology RD Company Limited
Queensland University of Technology
Type: Article
Journal: Computer Methods in Applied Mechanics and Engineering
Volume: 433
No. of pages: 37
ISSN: 0045-7825
Publication date: 01.01.2025
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: Computational Mechanics, Mechanics of Materials, Mechanical Engineering, General Physics and Astronomy, Computer Science Applications
Electronic version(s): https://doi.org/10.48550/arXiv.2406.11045 (Access: Open)
https://doi.org/10.1016/j.cma.2024.117518 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{19ffe52fbb0c45b68ae0935a47e0a3c8,
title = "Kolmogorov–Arnold-Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov–Arnold Networks",
abstract = "AI for partial differential equations (PDEs) has garnered significant attention, particularly with the emergence of Physics-informed neural networks (PINNs). The recent advent of Kolmogorov–Arnold Network (KAN) indicates that there is potential to revisit and enhance the previously MLP-based PINNs. Compared to MLPs, KANs offer interpretability and require fewer parameters. PDEs can be described in various forms, such as strong form, energy form, and inverse form. While mathematically equivalent, these forms are not computationally equivalent, making the exploration of different PDE formulations significant in computational physics. Thus, we propose different PDE forms based on KAN instead of MLP, termed Kolmogorov–Arnold-Informed Neural Network (KINN) for solving forward and inverse problems. We systematically compare MLP and KAN in various numerical examples of PDEs, including multi-scale, singularity, stress concentration, nonlinear hyperelasticity, heterogeneous, and complex geometry problems. Our results demonstrate that KINN significantly outperforms MLP regarding accuracy and convergence speed for numerous PDEs in computational solid mechanics, except for the complex geometry problem. This highlights KINN's potential for more efficient and accurate PDE solutions in AI for PDEs.",
keywords = "AI for PDEs, AI for science, Computational mechanics, Kolmogorov–Arnold Networks, PINNs",
author = "Yizheng Wang and Jia Sun and Jinshuai Bai and Cosmin Anitescu and Eshaghi, {Mohammad Sadegh} and Xiaoying Zhuang and Timon Rabczuk and Yinghua Liu",
note = "Publisher Copyright: {\textcopyright} 2024",
year = "2025",
month = jan,
day = "1",
doi = "10.48550/arXiv.2406.11045",
language = "English",
volume = "433",
journal = "Computer Methods in Applied Mechanics and Engineering",
issn = "0045-7825",
publisher = "Elsevier BV",
}

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