标题： KKANs：Kurkova-Kolmogorov-Arnold 网络及其学习动力学 显示英文标题

标题： KKANs: Kurkova-Kolmogorov-Arnold Networks and Their Learning Dynamics

摘要： 受Kolmogorov-Arnold表示定理和Kurkova使用近似表示原则的启发，我们提出了Kurkova-Kolmogorov-Arnold网络（KKAN），这是一种新的两块结构，将基于多层感知器（MLP）的稳健内部函数与灵活的基函数线性组合作为外部函数相结合。我们首先证明了KKAN是一个通用逼近器，然后展示了它在科学机器学习应用中的多功能性，包括函数回归、物理信息机器学习（PIML）和算子学习框架。基准结果表明，在函数逼近和算子学习任务中，KKAN优于MLP和原始的Kolmogorov-Arnold网络（KAN），并且在PIML中达到与完全优化的MLP相当的性能。为了更好地理解新表示模型的行为，我们使用信息瓶颈理论分析了它们的几何复杂性和学习动力学，识别出所有类型架构中的三个通用学习阶段：拟合、过渡和扩散。我们发现几何复杂性和信噪比（SNR）之间存在强烈相关性，最佳泛化在扩散阶段实现。此外，我们提出了基于自缩放残差的注意力权重，以动态保持高SNR，确保均匀收敛和延长学习。 显示英文摘要

摘要： Inspired by the Kolmogorov-Arnold representation theorem and Kurkova's principle of using approximate representations, we propose the Kurkova-Kolmogorov-Arnold Network (KKAN), a new two-block architecture that combines robust multi-layer perceptron (MLP) based inner functions with flexible linear combinations of basis functions as outer functions. We first prove that KKAN is a universal approximator, and then we demonstrate its versatility across scientific machine-learning applications, including function regression, physics-informed machine learning (PIML), and operator-learning frameworks. The benchmark results show that KKANs outperform MLPs and the original Kolmogorov-Arnold Networks (KANs) in function approximation and operator learning tasks and achieve performance comparable to fully optimized MLPs for PIML. To better understand the behavior of the new representation models, we analyze their geometric complexity and learning dynamics using information bottleneck theory, identifying three universal learning stages, fitting, transition, and diffusion, across all types of architectures. We find a strong correlation between geometric complexity and signal-to-noise ratio (SNR), with optimal generalization achieved during the diffusion stage. Additionally, we propose self-scaled residual-based attention weights to maintain high SNR dynamically, ensuring uniform convergence and prolonged learning.