基于模型与样本高效的人工智能辅助球体堆积数学发现

摘要

球体堆积问题，即希尔伯特第十八问题，探究n维欧几里得空间中全等球体的最密排列方式。尽管该问题与密码学、晶体学和医学成像等领域相关，但其研究仍悬而未决：除少数特殊维度外，既未找到最优堆积方案，也未能确定紧致上界。即使在n=8维度取得重大突破（该成果后来荣获菲尔兹奖），也凸显了其求解难度。求解上界的主流技术——三点法将该问题转化为求解大规模高精度半定规划（SDP）。由于每个候选SDP可能需要数日才能完成评估，传统数据密集型AI方法难以适用。我们通过将SDP构建建模为序贯决策过程（即SDP博弈），使策略能够从可容许构件集合中组装SDP公式，从而应对这一挑战。采用结合贝叶斯优化与蒙特卡洛树搜索的样本高效模型化框架，我们获得了维度4-16的最新上界结果，表明基于模型的搜索能推动长期几何问题的计算进展。这些成果共同证明，样本高效的模型化搜索能在数学严谨、评估受限的问题上取得实质性突破，为超越大规模LLM驱动探索的AI辅助发现指明了新方向。

English

Sphere packing, Hilbert's eighteenth problem, asks for the densest arrangement of congruent spheres in n-dimensional Euclidean space. Although relevant to areas such as cryptography, crystallography, and medical imaging, the problem remains unresolved: beyond a few special dimensions, neither optimal packings nor tight upper bounds are known. Even a major breakthrough in dimension n=8, later recognised with a Fields Medal, underscores its difficulty. A leading technique for upper bounds, the three-point method, reduces the problem to solving large, high-precision semidefinite programs (SDPs). Because each candidate SDP may take days to evaluate, standard data-intensive AI approaches are infeasible. We address this challenge by formulating SDP construction as a sequential decision process, the SDP game, in which a policy assembles SDP formulations from a set of admissible components. Using a sample-efficient model-based framework that combines Bayesian optimisation with Monte Carlo Tree Search, we obtain new state-of-the-art upper bounds in dimensions 4-16, showing that model-based search can advance computational progress in longstanding geometric problems. Together, these results demonstrate that sample-efficient, model-based search can make tangible progress on mathematically rigid, evaluation limited problems, pointing towards a complementary direction for AI-assisted discovery beyond large-scale LLM-driven exploration.

基于模型与样本高效的人工智能辅助球体堆积数学发现

Model-Based and Sample-Efficient AI-Assisted Math Discovery in Sphere Packing

摘要

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