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

TL;DR

This paper introduces a model-based, sample-efficient AI approach to improve upper bounds in sphere packing problems, demonstrating progress in dimensions 4-16 through a novel decision process and search framework.

Contribution

It formulates SDP construction as a sequential decision process and applies Bayesian optimization with Monte Carlo Tree Search to achieve new bounds in sphere packing.

Findings

Achieved new state-of-the-art upper bounds in dimensions 4-16.

Demonstrated the effectiveness of model-based search in mathematically rigid problems.

Showed that AI can make tangible progress on evaluation-limited geometric problems.

Abstract

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.