Algebraically-Informed Deep Networks (AIDN): A Deep Learning Approach to Represent Algebraic Structures

TL;DR

This paper introduces AIDN, a deep learning framework that encodes algebraic structures into neural networks, enabling automatic representation and analysis of complex algebraic and geometric objects relevant in topology.

Contribution

The paper presents a novel deep learning method that constructs algebraically-informed neural networks representing finitely-presented algebraic structures, bridging deep learning and algebra.

Findings

Successfully represents groups, algebras, and Lie algebras with neural networks.

Applies AIDN to solve Yang-Baxter equations and study braid groups.

Constructs new link invariants using deep learning techniques.

Abstract

One of the central problems in the interface of deep learning and mathematics is that of building learning systems that can automatically uncover underlying mathematical laws from observed data. In this work, we make one step towards building a bridge between algebraic structures and deep learning, and introduce \textbf{AIDN}, \textit{Algebraically-Informed Deep Networks}. \textbf{AIDN} is a deep learning algorithm to represent any finitely-presented algebraic object with a set of deep neural networks. The deep networks obtained via \textbf{AIDN} are \textit{algebraically-informed} in the sense that they satisfy the algebraic relations of the presentation of the algebraic structure that serves as the input to the algorithm. Our proposed network can robustly compute linear and non-linear representations of most finitely-presented algebraic structures such as groups, associative algebras, and Lie algebras. We evaluate our proposed approach and demonstrate its applicability to algebraic and geometric objects that are significant in low-dimensional topology. In particular, we study solutions for the Yang-Baxter equations and their applications on braid groups. Further, we study the representations of the Temperley-Lieb algebra. Finally, we show, using the Reshetikhin-Turaev construction, how our proposed deep learning approach can be utilized to construct new link invariants. We believe the proposed approach would tread a path toward a promising future research in deep learning applied to algebraic and geometric structures.