用机器学习识别黑洞熵的模对称结构，为量子引力找规律。
Machine learning automorphic forms for black holes
- 用神经网络从傅里叶系数反推模形式权重
- 负权模形式预测准确率高，尤其对应精确黑洞计数公式
- 适合研究量子引力对称性的学者，可自动发现隐藏规律
模形式、雅可比形式和拟模形式是生成BPS黑洞简并度的母函数。通过训练前馈神经网络，基于戴德金Eta函数、埃森斯坦级数及雅可比theta函数导出的模形式傅里叶系数，我们证明机器学习可准确预测截断展开中的模权重。结果显示，负权重的模形式与准模形式表现优异，特别是那些出现在精确黑洞计数公式中的情形；而正权重及更复杂的雅可比theta函数组合预测精度较低。本研究为利用机器学习识别引力系统中数据的模对称性提供了概念验证，并指明了在量子引力中自动化检测与验证对称性的路径。
原文摘要 · Abstract (English)
Modular, Jacobi, and mock-modular forms serve as generating functions for BPS black hole degeneracies. By training feed-forward neural networks on Fourier coefficients of automorphic forms derived from the Dedekind eta function, Eisenstein series, and Jacobi theta functions, we demonstrate that machine learning techniques can accurately predict modular weights from truncated expansions. Our results reveal strong performance for negative weight modular and quasi-modular forms, particularly those arising in exact black hole counting formulae, with lower accuracy for positive weights and more complicated combinations of Jacobi theta functions. This study establishes a proof of concept for using machine learning to identify how data is organized in terms of modular symmetries in gravitational systems and suggests a pathway toward automated detection and verification of symmetries in quantum gravity.
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