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.

翻译：偏微分方程的人工智能求解方法已引起广泛关注，尤其是在物理信息神经网络出现之后。最近提出的Kolmogorov-Arnold网络表明，重新审视并改进先前基于多层感知机的物理信息神经网络具有巨大潜力。与多层感知机相比，KAN具有更好的可解释性且所需参数更少。偏微分方程可用多种形式描述，如强形式、能量形式和反形式。这些形式在数学上等价，但在计算上并不等效，因此探索不同的偏微分方程表述形式在计算物理学中具有重要意义。为此，我们提出了基于KAN而非多层感知机的不同偏微分方程形式，称为Kolmogorov-Arnold信息神经网络。我们通过多尺度、奇异性、应力集中、非线性超弹性、非均匀材料和复杂几何等一系列偏微分方程数值算例，系统比较了多层感知机与KAN的性能。结果表明，除复杂几何问题外，KINN在计算固体力学众多偏微分方程求解的精度和收敛速度方面均显著优于多层感知机。这凸显了KINN在偏微分方程人工智能求解领域实现更高效、更精确求解的潜力。