几何柯尔莫哥洛夫-阿诺德叠加定理
Geometric Kolmogorov-Arnold Superposition Theorem
AI总结：
针对KAN难以建模具备几何对称性的物理系统的问题，提出融入多种群作用下等变性与不变性的KAT及KAN扩展方法，填补理论与架构空白，并通过分子动力学和粒子物理实验验证了有效性。
摘要：
柯尔莫哥洛夫-阿诺德定理（KAT），或更广义的柯尔莫哥洛夫叠加定理（KST），证明了任意非线性多元函数都可精确表示为非线性一元函数的有限叠加。与仅提供近似表示、不保证固定网络规模的通用逼近定理不同，KST提供了理论上精确的分解。柯尔莫哥洛夫-阿诺德网络（KAN）作为实现KAT的可训练模型被提出，近期进展结合现代神经网络概念对KAN进行了改进。然而，KAN难以有效建模需要E(3)变换等内在等变性或不变性几何对称性的物理系统，而这一性质对许多科学和工程应用至关重要。在本研究中，我们提出KAT和KAN的一种新型扩展，使其融入O(n)、O(1,n)、S_n及一般GL等多种群作用下的等变性与不变性，从而能够准确高效地对这类系统建模。我们的方法提供了一个统一框架，填补了物理系统的数学理论与实际架构之间的空白，将KAN的适用性拓展到更广泛的问题类别。我们在分子动力学系统和粒子物理领域开展了实验验证。
英文摘要：
The Kolmogorov-Arnold Theorem (KAT), or more generally, the Kolmogorov Superposition Theorem (KST), establishes that any non-linear multivariate function can be exactly represented as a finite superposition of non-linear univariate functions. Unlike the universal approximation theorem, which provides only an approximate representation without guaranteeing a fixed network size, KST offers a theoretically exact decomposition. The Kolmogorov-Arnold Network (KAN) was introduced as a trainable model to implement KAT, and recent advancements have adapted KAN using concepts from modern neural networks. However, KAN struggles to effectively model physical systems that require inherent equivariance or invariance geometric symmetries as $E(3)$ transformations, a key property for many scientific and engineering applications. In this work, we propose a novel extension of KAT and KAN to incorporate equivariance and invariance over various group actions, including $O(n)$, $O(1,n)$, $S_n$, and general $GL$, enabling accurate and efficient modeling of these systems. Our approach provides a unified approach that bridges the gap between mathematical theory and practical architectures for physical systems, expanding the applicability of KAN to a broader class of problems. We provide experimental validation on molecular dynamical systems and particle physics.
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