Geometric Kolmogorov-Arnold Superposition Theorem

2025-02-23

AI Summary

Kolmogorov–Arnold Networks (KANs) lack inherent $E(3)$-equivariance/invariance, limiting their applicability to physical systems governed by Euclidean symmetries. Method: This work generalizes the Kolmogorov–Arnold theorem to geometric settings, establishing an $O(n)$-equivariant geometric Kolmogorov–Arnold Theorem (KAT) framework. We propose the first strictly $SE(3)$-equivariant KAN architecture: it employs learnable equivariant activation functions grounded in group representation theory and differential geometry, integrated with piecewise-smooth univariate mappings and $SE(3)$-equivariant coordinate encoding. Contribution/Results: Our model achieves state-of-the-art performance on molecular energy prediction and particle dynamics modeling, reducing parameter count by over 40% versus prior equivariant models. Crucially, it provides theoretical guarantees of exact $SE(3)$-equivariance—under arbitrary rotations and translations—and universal approximation capability for $SE(3)$-invariant/equivariant functions.

Technology Category

• Machine Learning: Learning with Manifolds
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Computer Vision: Generative Adversarial Networks (GANs) for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

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 to $E(3)$ 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 $O(n)$ group actions, 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.

Problem

• Extends Kolmogorov-Arnold Network
• Incorporates equivariance and invariance
• Models physical systems accurately

Innovation

• Extends Kolmogorov-Arnold Network (KAN)
• Incorporates O(n) group actions
• Enhances modeling of physical systems