本文对 18 种不同的多项式及其在 Kolmogorov-Arnold Network (KAN) 模型中作为传统基于样条方法替代方案的潜在应用进行了全面调查。根据数学性质，这些多项式被分为正交多项式、超几何多项式、q-多项式、与 Fibonacci 相关的多项式、组合多项式和数论多项式等不同组。本研究旨在调查这些多项式作为 KAN 模型中基函数的适用性，用于处理 MNIST 数据集上的手写数字分类等复杂任务。对 KAN 模型的性能指标（包括总体准确率、Kappa 和 F1 分数）进行了评估和比较。Gottlieb-KAN 模型在所有指标上均取得了最高性能，表明其作为该给定任务合适选择的潜力。然而，为了充分理解它们在 KAN 模型中的能力，有必要在更复杂的数据集上对这些多项式进行进一步的分析和调优。这些 KAN 模型实现的源代码可在 https://github.com/seydi1370/Basis_Functions 获取。
英文摘要：
This paper presents a comprehensive survey of 18 distinct polynomials and their potential applications in Kolmogorov-Arnold Network (KAN) models as an alternative to traditional spline-based methods. The polynomials are classified into various groups based on their mathematical properties, such as orthogonal polynomials, hypergeometric polynomials, q-polynomials, Fibonacci-related polynomials, combinatorial polynomials, and number-theoretic polynomials. The study aims to investigate the suitability of these polynomials as basis functions in KAN models for complex tasks like handwritten digit classification on the MNIST dataset. The performance metrics of the KAN models, including overall accuracy, Kappa, and F1 score, are evaluated and compared. The Gottlieb-KAN model achieves the highest performance across all metrics, suggesting its potential as a suitable choice for the given task. However, further analysis and tuning of these polynomials on more complex datasets are necessary to fully understand their capabilities in KAN models. The source code for the implementation of these KAN models is available at https://github.com/seydi1370/Basis_Functions .
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Department of Computer, Control and Management Engineering, Sapienza University of Rome（西帕辛扎大学罗马分校计算机、控制与管理工程系）；Department of Life Science Informatics and Data Science, B-IT, LIMES Program Unit Chemical Biology and Medicinal Chemistry, Rheinische Friedrich-Wilhelms-Universität, Bonn, Germany（波恩德国莱茵-威斯巴登大学生命科学信息技术与数据科学系）；Lamarr Institute for Machine Learning and Artificial Intelligence, Bonn, Germany（波恩德国拉马尔机器学习与人工智能研究所）
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University of Bath（巴斯大学）；Zhejiang University（浙江大学）
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Key Laboratory of Universal Wireless Communication, Ministry of Education, School of Information and Communication Engineering, Beijing University of Posts and Telecommunications（信息与通信工程学院，北京邮电大学，教育部无线通信重点实验室）；Department of Informatics and Telecommunications, National and Kapodistrian University of Athens（信息与电信学院，希腊国家与卡波迪斯蒂亚纳大学）；telecommunication company in Stockholm, Sweden（瑞典斯德哥尔摩的一家电信公司）；School of Computing, Engineering and Mathematical Sciences, La Trobe University（计算、工程与数学科学学院，拉特罗布大学）
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Department of Mathematics, Hong Kong Baptist University（香港 Baptist大学数学系）；Institution for Foundations of Data Science, Yale University（数据科学基础研究所，耶鲁大学）；Robotics and Technology of Computers Lab., Universidad de Sevilla（机器人与计算机技术实验室，塞维利亚大学）
