用强化学习在高维格点中找新代数几何对象，发现上千个未知的4维法诺超曲面。
Deep Reinforcement Learning for Fano Hypersurfaces
- 用深度强化学习设计动态搜索启发式，引导探索密集奖励区域。
- 找到数千个此前未知的带终端奇点的4维法诺超曲面，数百个无法被传统方法发现。
- 适合对代数几何、智能搜索或高维组合优化感兴趣的读者。
我们设计了一种深度强化学习算法，用于在高维整数格点中进行稀疏奖励环境下的探索，通过训练前馈神经网络作为动态搜索启发式，引导探索向奖励密集区域。该方法应用于发现带有终端奇点的4维法诺超曲面，这类对象在代数几何中具有核心地位。法诺簇是代数簇的基本构建块，其显式例子对理论发展与推广至关重要。尽管历经数十年研究，由于底层搜索空间的组合难题，该分类仍严重不完整。我们的强化学习方法发现了数千个此前未知的例子，其中数百个已被证明无法通过已知搜索方法获取。
原文摘要 · Abstract (English)
We design a deep reinforcement learning algorithm to explore a high-dimensional integer lattice with sparse rewards, training a feedforward neural network as a dynamic search heuristic to steer exploration toward reward dense regions. We apply this to the discovery of Fano 4-fold hypersurfaces with terminal singularities, objects of central importance in algebraic geometry. Fano varieties with terminal singularities are fundamental building blocks of algebraic varieties, and explicit examples serve as a vital testing ground for the development and generalisation of theory. Despite decades of effort, the combinatorial intractability of the underlying search space has left this classification severely incomplete. Our reinforcement learning approach yields thousands of previously unknown examples, hundreds of which we show are inaccessible to known search methods.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。