asKAN：嵌入主动子空间的Kolmogorov-Arnold网络
asKAN: Active Subspace embedded Kolmogorov-Arnold Network
AI总结：
针对KAN难以灵活建模脊函数的问题，提出asKAN框架，通过在KAN之间嵌入主动子空间检测来识别脊方向并自适应投影自变量，在不增加神经元数量的情况下显著降低函数拟合、Poisson方程求解和声场重建误差。
摘要：
Kolmogorov-Arnold网络（KAN）已成为面向小规模AI+科学应用的一种有前景的神经网络架构。然而，它在建模脊函数方面缺乏灵活性，而脊函数广泛用于表示物理系统中的关系。本研究通过Kolmogorov-Arnold定理的视角考察这一不灵活性，该定理从构造单变量分量而非组合自变量开始表示多元函数。我们的分析表明，引入自变量的线性组合可以显著简化表示脊函数时的网络架构。受这一发现启发，我们提出主动子空间嵌入KAN（asKAN），这是一个将KAN的函数表示能力与主动子空间方法协同结合的层次化框架。该架构在KAN之间策略性地嵌入主动子空间检测，其中主动子空间方法用于识别主要脊方向，并将自变量自适应投影到这些关键维度上。所提出的asKAN以迭代方式实现，且不增加原始KAN中的神经元数量。所提方法通过函数拟合、求解Poisson方程以及声场重建进行了验证。与KAN相比，asKAN在相同网络架构下显著降低了误差。结果表明，asKAN增强了KAN在拟合和求解脊函数形式方程方面的能力。
英文摘要：
The Kolmogorov-Arnold Network (KAN) has emerged as a promising neural network architecture for small-scale AI+Science applications. However, it suffers from inflexibility in modeling ridge functions, which is widely used in representing the relationships in physical systems. This study investigates this inflexibility through the lens of the Kolmogorov-Arnold theorem, which starts the representation of multivariate functions from constructing the univariate components rather than combining the independent variables. Our analysis reveals that incorporating linear combinations of independent variables can substantially simplify the network architecture in representing the ridge functions. Inspired by this finding, we propose active subspace embedded KAN (asKAN), a hierarchical framework that synergizes KAN's function representation with active subspace methodology. The architecture strategically embeds active subspace detection between KANs, where the active subspace method is used to identify the primary ridge directions and the independent variables are adaptively projected onto these critical dimensions. The proposed asKAN is implemented in an iterative way without increasing the number of neurons in the original KAN. The proposed method is validated through function fitting, solving the Poisson equation, and reconstructing sound field. Compared with KAN, asKAN significantly reduces the error using the same network architecture. The results suggest that asKAN enhances the capability of KAN in fitting and solving equations in the form of ridge functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
