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

Research output: Contribution to journalArticleResearchpeer review

Authors

  • Yizheng Wang
  • Jia Sun
  • Jinshuai Bai
  • Cosmin Anitescu
  • Mohammad Sadegh Eshaghi
  • Xiaoying Zhuang
  • Timon Rabczuk
  • Yinghua Liu

Research Organisations

External Research Organisations

  • Tsinghua University
  • Bauhaus-Universität Weimar
  • CNPC Engineering Technology RD Company Limited
  • Queensland University of Technology
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Details

Original languageEnglish
Article number117518
Number of pages37
JournalComputer Methods in Applied Mechanics and Engineering
Volume433
Early online date18 Nov 2024
Publication statusPublished - 1 Jan 2025

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

ASJC Scopus subject areas

Cite this

Kolmogorov–Arnold-Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov–Arnold Networks. / Wang, Yizheng; Sun, Jia; Bai, Jinshuai et al.
In: Computer Methods in Applied Mechanics and Engineering, Vol. 433, 117518, 01.01.2025.

Research output: Contribution to journalArticleResearchpeer review

Wang Y, Sun J, Bai J, Anitescu C, Eshaghi MS, Zhuang X et al. Kolmogorov–Arnold-Informed neural network: A physics-informed deep learning framework for solving forward and inverse problems based on Kolmogorov–Arnold Networks. Computer Methods in Applied Mechanics and Engineering. 2025 Jan 1;433:117518. Epub 2024 Nov 18. doi: 10.48550/arXiv.2406.11045, 10.1016/j.cma.2024.117518
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AU - Zhuang, Xiaoying

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