Machine learning modularity

TL;DR

This work addresses the problem of obtaining symbolic simplifications governed by modular identities for special functions, notably the $q$-$\theta$ function and the elliptic Gamma function. It proposes a transformer‑based framework with dynamic batching to learn and apply $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z})$ modular transformations, enabling automatic simplification of heavily scrambled expressions. The study demonstrates robust generalization: near‑perfect accuracy on in‑distribution tests and strong extrapolation performance under deep scrambling, indicating that the models capture underlying algebraic rules rather than mere memorization. These results open the way for automated symbolic tools in quantum field theory and string theory computations, with potential extensions to higher‑rank modularities and hybrid expressions involving polylogarithms.

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 $q$-$θ$ 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.

Figures (7)

• Figure 1: Hyperbolic cutoff radius $R_{h} = 5$, Euclidean radius $r = 0.99$, $N = 20{,}000$.
• Figure 2: Bertrand I sampling.
• Figure 3: Bertrand I (chord-midpoint) sampling, $N = 20{,}000$.
• Figure 4: Bertrand II (Euclidean area-uniform) sampling, $N = 20{,}000$.
• Figure 5: Bertrand III (random radius midpoint) sampling, $N = 20{,}000$.