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.

该论文通过机器学习辅助框架，对仿射Deligne-Lusztig变量的几何形态展开了新颖的、跨学科的研究。研究的主要目标是调查不可约分量的非空模式、维度和枚举。作者提出的框架展示了数据生成、模型训练、模式分析和人工审查之间错综复杂的相互作用，强调数据生成过程的细微之处，突出了有意义的子集和适当的特征集的选择。作者表明该框架对加速纯数学研究具有潜力，并能导致新的猜想和有前途的研究方向的发现。作者还分享了计算仿射Deligne-Lusztig变量和机器学习模型的源代码，促进进一步的探索，并总结了这次合作的宝贵经验和教训。

基于机器学习辅助的仿射 Deligne-Lusztig 多样体探索