This paper presents a novel, interdisciplinary study that leverages a Machine Learning (ML) assisted framework to explore the geometry of affine Deligne-Lusztig varieties (ADLV). The primary objective is to investigate the nonemptiness pattern, dimension and enumeration of irreducible components of ADLV. Our proposed framework demonstrates a recursive pipeline of data generation, model training, pattern analysis, and human examination, presenting an intricate interplay between ML and pure mathematical research. Notably, our data-generation process is nuanced, emphasizing the selection of meaningful subsets and appropriate feature sets. We demonstrate that this framework has a potential to accelerate pure mathematical research, leading to the discovery of new conjectures and promising research directions that could otherwise take significant time to uncover. We rediscover the virtual dimension formula and provide a full mathematical proof of a newly identified problem concerning a certain lower bound of dimension. Furthermore, we extend an open invitation to the readers by providing the source code for computing ADLV and the ML models, promoting further explorations. This paper concludes by sharing valuable experiences and highlighting lessons learned from this collaboration.

翻译：本文提出了一项新颖的跨学科研究，利用机器学习（ML）辅助框架探索仿射Deligne-Lusztig簇（ADLV）的几何性质。主要目标是研究ADLV的非空性模式、维数以及不可约分支的枚举。我们的框架构建了一个递归流程，涵盖数据生成、模型训练、模式分析及人工核查，展现了机器学习与纯数学研究之间的复杂相互作用。值得注意的是，数据生成过程精细严谨，强调对有意义子集及特征集的选取。研究表明，该框架具有加速纯数学研究的潜力，能够发现原本需要大量时间才能揭示的新猜想和有前景的研究方向。我们重新发现了虚维数公式，并为新识别出的关于维数下界的问题提供了完整的数学证明。此外，我们向读者开放了计算ADLV的源代码及ML模型，以促进进一步探索。本文最后分享了从此次合作中获得的宝贵经验与深刻启示。