Machine learning modularity

TL;DR

This paper introduces a transformer-based machine learning framework that automatically simplifies complex elliptic Gamma function expressions by learning algebraic identities, achieving high accuracy and robustness in symbolic simplification tasks.

Contribution

It presents the first successful application of machine learning to symbolic simplification using modular identities, advancing automated computation in special functions.

Findings

Over 99% accuracy on in-distribution tests

Robust performance exceeding 90% under extrapolation

Model internalizes algebraic rules rather than memorizing

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.