Machine learning modularity

2026-01-05

AI Summary

This work proposes the first framework that applies machine learning to symbolic simplification based on modular identities, specifically targeting complex expressions involving elliptic Gamma functions (including q-theta functions). The approach leverages a Transformer-based sequence-to-sequence architecture enhanced with dynamic batching and explicit encoding of algebraic rules derived from SL(2,ℤ) and SL(3,ℤ) modular transformations. This design enables the model to internalize the underlying algebraic structure rather than merely memorizing superficial patterns. Experimental results demonstrate that the method achieves over 99% accuracy on in-distribution test cases and maintains above 90% accuracy under deep extrapolation scenarios, reflecting both strong generalization capabilities and a profound understanding of modular transformation rules.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Based on a transformer based sequence-to-sequence architecture combined with a dynamic batching algorithm, this work introduces a machine learning framework for automatically simplifying complex expressions involving multiple elliptic Gamma functions, including the $q$-$\theta$ function and the elliptic Gamma function. The model learns to apply algebraic identities, particularly the SL$(2,\mathbb{Z})$ and SL$(3,\mathbb{Z})$ modular transformations, to reduce heavily scrambled expressions to their canonical forms. Experimental results show that the model achieves over 99\% accuracy on in-distribution tests and maintains robust performance (exceeding 90\% accuracy) under significant extrapolation, such as with deeper scrambling depths. This demonstrates that the model has internalized the underlying algebraic rules of modular transformations rather than merely memorizing training patterns. Our work presents the first successful application of machine learning to perform symbolic simplification using modular identities, offering a new automated tool for computations with special functions in quantum field theory and the string theory.

Problem

• elliptic Gamma function
• modular transformations
• symbolic simplification
• special functions
• q-θ function

Innovation

• modular transformations
• elliptic Gamma function
• symbolic simplification
• transformer architecture
• dynamic batching