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.

翻译：基于Transformer序列到序列架构并结合动态批处理算法，本研究提出了一种机器学习框架，用于自动简化涉及多个椭圆Gamma函数（包括$q$-$θ$函数和椭圆Gamma函数）的复杂表达式。该模型学习应用代数恒等式，特别是SL$(2,\mathbb{Z})$和SL$(3,\mathbb{Z})$模变换，将高度扰乱的表达式约化为规范形式。实验结果表明，该模型在分布内测试中达到超过99%的准确率，并在显著外推（如更深扰乱深度）下保持稳健性能（准确率超过90%）。这表明模型已内化模变换的基本代数规则，而非仅仅记忆训练模式。我们的工作首次成功应用机器学习执行基于模恒等式的符号简化，为量子场论和弦理论中的特殊函数计算提供了新的自动化工具。