arXiv 2406.110452024-08-04cs.LGcs.NAmath.NA

Kolmogorov-Arnold知情神经网络：一种基于Kolmogorov-Arnold网络求解正问题和反问题的物理知情深度学习框架

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

Tsinghua University（清华大学）；Bauhaus-Universität Weimar（魏玛包豪斯大学）；CNPC Engineering Technology RD Company Limited（中国石油集团工程技术研究院有限公司）；Queensland University of Technology（昆士兰科技大学）；Leibniz University Hannover（汉诺威莱布尼茨大学）

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

发表信息：: Comput. Methods Appl. Mech. Engrg. 433 (2025) 117518
DOI：: 10.1016/j.cma.2024.117518

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AI总结：

本文提出基于KAN的物理知情框架KINN，用于求解PDE正问题和反问题，并在多种固体力学算例中证明其在精度和收敛速度上显著优于基于MLP的PINNs。

摘要：

面向偏微分方程（PDEs）的人工智能已引起广泛关注，尤其是随着物理知情神经网络（PINNs）的出现。近期Kolmogorov-Arnold网络（KAN）的提出表明，重新审视并增强此前基于多层感知机（MLP）的PINNs存在潜力。与MLP相比，KAN具有可解释性且所需参数更少。偏微分方程可以用多种形式描述，例如强形式、能量形式和反问题形式。尽管这些形式在数学上等价，但在计算上并不等价，因此探索不同的PDE表述在计算物理中具有重要意义。为此，我们提出了基于KAN而非MLP的不同PDE形式，称为Kolmogorov-Arnold知情神经网络（KINN），用于求解正问题和反问题。我们在多个PDE数值算例中系统比较了MLP和KAN，包括多尺度、奇异性、应力集中、非线性超弹性、非均质以及复杂几何问题。结果表明，除复杂几何问题外，KINN在计算固体力学的大量PDE中在精度和收敛速度方面显著优于MLP。这凸显了KINN在面向PDE的人工智能中实现更高效、更精确PDE求解的潜力。

英文摘要：

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) 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.

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