Geometric Kolmogorov-Arnold Superposition Theorem

TL;DR

This paper extends the Kolmogorov-Arnold Superposition Theorem and its neural network implementation to incorporate geometric symmetries, enabling precise modeling of physical systems with invariance and equivariance properties.

Contribution

It introduces a novel extension of KAT and KAN to include invariance and equivariance over various groups, broadening their applicability to physical systems.

Findings

Effective modeling of molecular dynamics systems

Successful incorporation of geometric symmetries

Enhanced neural network architectures for physical invariance

Abstract

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.