用新型神经网络解决高指标微分代数方程，精度提升1到2个数量级。
DAE-KAN: A Kolmogorov-Arnold Network Model for High-Index Differential-Algebraic Equations
- 将Kolmogorov-Arnold网络与物理信息神经网络结合，提升函数拟合能力。
- 对1至3阶指标的DAE系统，误差降低1~2个数量级，显著优于传统PINN。
- 适合求解复杂物理系统的微分代数方程，尤其在精度和泛化上表现优异。
Kolmogorov-Arnold网络（KANs）因其在数据驱动建模中卓越的函数拟合能力，成为多层感知机（MLPs）的有前景替代方案。本文提出一种新框架DAE-KAN，通过将KANs与物理信息神经网络（PINNs）结合，用于求解高指标微分代数方程（DAEs）。该框架不仅保持了传统PINNs对物理规律支配复杂系统的建模能力，还通过KAN的函数拟合优势提升了性能。数值实验表明，对于从1阶到3阶指标的DAE系统，DAE-KAN使微分变量和代数变量的绝对误差降低1至2个数量级，相比传统PINNs有显著改进。为评估效果，我们分析了漂移误差，发现DAE-KAN与PINNs均优于经典数值方法。结果表明，神经网络方法，尤其是DAE-KAN，具有高计算精度与良好泛化能力，为挑战性偏微分代数方程提供了有效解决方案。
原文摘要 · Abstract (English)
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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