DAE-KAN：一种用于高指标微分代数方程的Kolmogorov-Arnold网络模型
DAE-KAN: A Kolmogorov-Arnold Network Model for High-Index Differential-Algebraic Equations
AI总结：
本文提出DAE-KAN框架，将Kolmogorov-Arnold网络与物理信息神经网络结合，用于求解高指标微分代数方程，相比传统PINNs将误差降低1-2个数量级，并优于经典数值方法控制漂移误差。
摘要：
Kolmogorov-Arnold网络（KANs）因其在数据驱动建模中卓越的函数拟合能力，已成为多层感知器（MLPs）的有前景的替代方案。本文提出了一种新颖框架DAE-KAN，通过将KANs与物理信息神经网络（PINNs）相结合，用于求解高指标微分代数方程（DAEs）。该框架不仅保留了传统PINNs对受物理定律支配的复杂系统进行建模的能力，而且通过利用KANs的函数拟合优势增强了其性能。数值实验表明，对于从指标1到指标3的DAE系统，与传统PINNs相比，DAE-KAN将微分变量和代数变量的绝对误差降低了1到2个数量级。为了评估该方法的有效性，我们分析了漂移误差，发现PINNs和DAE-KAN在控制这一现象方面均优于经典数值方法。我们的结果凸显了神经网络方法，特别是DAE-KAN，在求解高指标DAEs方面的潜力，具有显著的计算精度和泛化能力，为具有挑战性的偏微分代数方程提供了一种有前景的解决方案。
英文摘要：
Kolmogorov-Arnold Networks (KANs) have emerged as a promising alternative to Multi-layer Perceptrons (MLPs) due to their superior function-fitting abilities in data-driven modeling. In this paper, we propose a novel framework, DAE-KAN, for solving high-index differential-algebraic equations (DAEs) by integrating KANs with Physics-Informed Neural Networks (PINNs). This framework not only preserves the ability of traditional PINNs to model complex systems governed by physical laws but also enhances their performance by leveraging the function-fitting strengths of KANs. Numerical experiments demonstrate that for DAE systems ranging from index-1 to index-3, DAE-KAN reduces the absolute errors of both differential and algebraic variables by 1 to 2 orders of magnitude compared to traditional PINNs. To assess the effectiveness of this approach, we analyze the drift-off error and find that both PINNs and DAE-KAN outperform classical numerical methods in controlling this phenomenon. Our results highlight the potential of neural network methods, particularly DAE-KAN, in solving high-index DAEs with substantial computational accuracy and generalization, offering a promising solution for challenging partial differential-algebraic equations.
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