基于Kolmogorov-Arnold网络的科学机器学习
Scientific Machine Learning with Kolmogorov-Arnold Networks
AI总结：
本综述全面回顾了Kolmogorov-Arnold Networks（KANs）在科学机器学习中的最新进展，从数据驱动、物理信息和深度算子学习三个视角对比了其相较于MLPs在准确性与可解释性上的优势，并探讨了当前挑战与未来方向。
摘要：
科学机器学习领域最初使用多层感知机（MLPs），如今正越来越多地采用Kolmogorov-Arnold Networks（KANs）进行数据编码。这一转变是由MLPs的局限性所推动的，包括可解释性差、激活函数固定，以及难以捕捉局部或高频特征。KANs通过增强的可解释性和灵活性解决了这些问题，能够更高效地对复杂的非线性交互进行建模，并有效克服与传统MLP架构相关的约束。本综述从三个不同视角对基于KAN的模型的最新进展进行了分类：(i) 数据驱动学习，(ii) 物理信息建模，以及(iii) 深度算子学习。每个视角都通过架构设计、训练策略、应用效果以及与基于MLP的对应模型进行比较评估的视角进行了考察。通过将KANs与MLPs进行基准测试，我们突出了在准确性、收敛性和频谱表示方面的持续改进，阐明了KANs在捕捉复杂动力学同时更有效学习方面的优势。除了回顾近期文献外，本工作还提出了几项比较评估，阐明了KAN建模的核心特征，并暗示了它们对现实世界应用的潜在影响。最后，本综述指出了KAN发展中的关键挑战和开放研究问题，特别是在计算效率、理论保证、超参数调优和算法复杂性方面。我们还概述了旨在提高基于KAN框架的鲁棒性、可扩展性和物理一致性的未来研究方向。
英文摘要：
The field of scientific machine learning, which originally utilized multilayer perceptrons (MLPs), is increasingly adopting Kolmogorov-Arnold Networks (KANs) for data encoding. This shift is driven by the limitations of MLPs, including poor interpretability, fixed activation functions, and difficulty capturing localized or high-frequency features. KANs address these issues with enhanced interpretability and flexibility, enabling more efficient modeling of complex nonlinear interactions and effectively overcoming the constraints associated with conventional MLP architectures. This review categorizes recent progress in KAN-based models across three distinct perspectives: (i) data-driven learning, (ii) physics-informed modeling, and (iii) deep-operator learning. Each perspective is examined through the lens of architectural design, training strategies, application efficacy, and comparative evaluation against MLP-based counterparts. By benchmarking KANs against MLPs, we highlight consistent improvements in accuracy, convergence, and spectral representation, clarifying KANs' advantages in capturing complex dynamics while learning more effectively. In addition to reviewing recent literature, this work also presents several comparative evaluations that clarify central characteristics of KAN modeling and hint at their potential implications for real-world applications. Finally, this review identifies critical challenges and open research questions in KAN development, particularly regarding computational efficiency, theoretical guarantees, hyperparameter tuning, and algorithm complexity. We also outline future research directions aimed at improving the robustness, scalability, and physical consistency of KAN-based frameworks.
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