用Sinc插值改进KAN网络，更好求解带奇点的偏微分方程
Sinc Kolmogorov-Arnold network and its application for solving PDEs with singularities
- 将Sinc插值引入KAN，让网络能自适应学习奇异函数
- 在多个含奇点的PDE问题上，性能优于传统方法
- 适合需要高精度处理不连续或奇异性问题的研究者
本文提出在柯尔莫哥洛夫-阿诺德网络（Kolmogorov-Arnold Networks, KAN）中使用Sinc插值，该类神经网络通过可学习的激活函数替代传统多层感知机。尽管已有多种函数表示方式被尝试，但本研究证明Sinc插值是一种可行替代方案，因其在数值分析中已被证实能有效逼近光滑函数与具有奇点的函数。这对函数逼近及物理信息神经网络求解偏微分方程尤为重要。通过一系列实验验证，SincKAN在几乎所有测试案例中均取得更优结果。
原文摘要 · Abstract (English)
In this paper, we propose to use Sinc interpolation in the context of Kolmogorov-Arnold Networks, neural networks with learnable activation functions, which recently gained attention as alternatives to Multilayer Perceptron. Many different function representations have already been tried, but we show that Sinc interpolation proposes a viable alternative, since it is known in numerical analysis to effectively represent both smooth functions and functions with singularities. This is important not only for function approximation but also for solving the partial differential equations with physics-informed neural networks. Through a series of experiments, we show that SincKANs provide better results in almost all of the examples we have considered.
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