Kolmogorov-Arnold Networks (KANs) have recently emerged as a promising alternative to traditional Multilayer Perceptrons (MLPs), inspired by the Kolmogorov-Arnold representation theorem. Unlike MLPs, which use fixed activation functions on nodes, KANs employ learnable univariate basis functions on edges, offering enhanced expressivity and interpretability. This review provides a systematic and comprehensive overview of the rapidly expanding KAN landscape, moving beyond simple performance comparisons to offer a structured synthesis of theoretical foundations, architectural variants, and practical implementation strategies. By collecting and categorizing a vast array of open-source implementations, we map the vibrant ecosystem supporting KAN development. We begin by bridging the conceptual gap between KANs and MLPs, establishing their formal equivalence and highlighting the superior parameter efficiency of the KAN formulation. A central theme of our review is the critical role of the basis function; we survey a wide array of choices, including B-splines, Chebyshev and Jacobi polynomials, ReLU compositions, Gaussian RBFs, and Fourier series, and analyze their respective trade-offs in terms of smoothness, locality, and computational cost. We then categorize recent advancements into a clear roadmap, covering techniques for improving accuracy, efficiency, and regularization. Key topics include physics-informed loss design, adaptive sampling, domain decomposition, hybrid architectures, and specialized methods for handling discontinuities. Finally, we provide a practical "Choose-Your-KAN" guide to help practitioners select appropriate architectures, and we conclude by identifying current research gaps. The associated GitHub repository https://github.com/AmirNoori68/kan-review complements this paper and serves as a structured reference for ongoing KAN research.

翻译：Kolmogorov-Arnold网络（KANs）作为传统多层感知机（MLPs）的一种有前景的替代方案，近期受到关注，其灵感来源于Kolmogorov-Arnold表示定理。与MLPs在节点上使用固定激活函数不同，KANs在边上采用可学习的单变量基函数，从而提供更强的表达能力和可解释性。本综述对快速发展的KAN领域进行了系统而全面的概述，超越简单的性能比较，提供了理论基础、架构变体和实际实施策略的结构化综合。通过收集和分类大量开源实现，我们描绘了支持KAN发展的活跃生态系统。我们首先弥合KANs与MLPs之间的概念差距，建立它们的正式等价性，并强调KAN公式在参数效率上的优越性。我们综述的核心主题是基函数的关键作用；我们调研了多种选择，包括B样条、切比雪夫和雅可比多项式、ReLU组合、高斯径向基函数和傅里叶级数，并分析了它们在平滑性、局部性和计算成本方面的权衡。随后，我们将近期进展分类为一个清晰的路线图，涵盖提高准确性、效率和正则化的技术。关键主题包括物理信息损失设计、自适应采样、域分解、混合架构以及处理间断性的专门方法。最后，我们提供了一个实用的“选择你的KAN”指南，以帮助实践者选择合适的架构，并通过识别当前研究空白作为总结。相关的GitHub仓库https://github.com/AmirNoori68/kan-review补充了本文，并作为持续KAN研究的结构化参考。