基于模型与样本高效的人工智能辅助球体堆叠数学发现

摘要

球體堆積問題——希爾伯特第十八問題，探究n維歐幾里得空間中全等球體的最密排列方式。儘管該問題與密碼學、晶體學和醫學成像等領域相關，其解決方案仍懸而未決：除少數特殊維度外，既未發現最優堆積結構，也未能建立緊緻的上界。即使在n=8維度取得並後獲菲爾茲獎認可的重大突破，亦凸顯了此問題的難度。現有主流的上界推導技術「三點定位法」，將問題轉化為求解大規模高精度半定規劃（SDP）問題。由於每個候選SDP的評估可能耗時數日，傳統數據密集型AI方法難以適用。我們通過將SDP構建建模為順序決策過程（即SDP博弈），提出解決方案：該策略從一組可容許組件中逐步組裝SDP公式。採用結合貝葉斯優化與蒙特卡洛樹搜索的樣本高效模型化框架，我們在4-16維空間中獲得了當前最優的上界結果，證明了基於模型的搜索能推動經典幾何問題的計算進展。這些成果共同表明，樣本高效的模型化搜索能在數學結構嚴謹、評估資源受限的問題上取得實質進展，為超越大規模語言模型驅動探索的AI輔助發現開辟了新的路徑。

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叠数学发现

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

摘要

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