人工神经网络
计算机科学
人工智能
统计物理学
数学教育
认知科学
应用数学
心理学
物理
数学
作者
Yizheng Wang,Jia Sun,Jinshuai Bai,Cosmin Anitescu,Mohammad Sadegh Es-haghi,Xiaoying Zhuang,Timon Rabczuk,Yinghua Liu
出处
期刊:Cornell University - arXiv
日期:2024-06-16
标识
DOI:10.48550/arxiv.2406.11045
摘要
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). 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 in terms of 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.
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