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

TL;DR

The paper tackles the challenging problem of tightly bounding sphere-packing densities by reframing SDP construction as a sequential, model-based game (the SDP Game). It couples Bayesian optimization over geometric parameters $(r,R)$ with Monte Carlo Tree Search over a token-based polynomial vocabulary to synthesize admissible three-point bound certificates, enabling sample-efficient exploration under expensive SDP evaluations. Across dimensions $4$–$16$, the approach yields new state-of-the-art upper bounds and uncovers substantial novel polynomial structure, while in the $n=8$ case it numerically approaches the known optimum without relying on modular-form inputs. This work demonstrates a principled AI-assisted pathway for solving evaluation-limited mathematical problems, offering a complementary avenue to large-scale LLM-driven exploration and suggesting broader applicability to other rigid analytic settings.

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.

Figures (7)

• Figure 1: A Brief History of the best known sphere-packing bounds across dimensions. This figure highlights the classical milestones in sphere-packing. The range $4 \leq n \leq 16$ has been the most intensely studied, with decades-long progress driven by lattice and non-lattice constructions (see $D_4$, $E_6$, $E_7$, $E_8$, $K_{12}$, Barnes-Wall, Leech, and others). Our model-based AI approach (blue stars, 2025) achieves new state-of-the-art bounds throughout this critical band of dimensions, surpassing all previously known constructions.
• Figure 2: Three-Point Bound Method for Sphere Packing Problems. This method establishes upper bounds on the density of arbitrary sphere packings in $\mathbb{R}^{\text{n}}$ by formulating specific semidefinite programming (SDP) problems. The process begins with the choice of two nonnegative geometric parameters $(r,R)$ with $R > r$. Given these parameters, one constructs the function class $\mathcal{F}(r,R)$. Each pair $(r, R)$ together with functions $f_1, f_2 \in \mathcal{F}(r, R)$ define a specific semidefinite program $\text{SDP}(r, R, f_1, f_2)$. These SDPs, are then solved using standard SDP solvers, and their optimal objective values provide valid upper bounds on the sphere-packing density.
• Figure 3: Overall Framework to Solve SDP Games. Our approach solves SDP Games through a four-stage, sample-efficient search loop. (1) The agent begins by using Bayesian optimisation to propose geometric parameters $(r,R)$, updating a surrogate model from past evaluations and trading off exploration versus exploitation. (2) Given this choice, it constructs the admissible vocabulary of polynomial building blocks associated with $\mathcal{F}(r,R)$. (3) Monte Carlo Tree Search then explores combinations of these building blocks to assemble candidate SDP formulations that are predicted to yield stronger bounds. (4) The resulting SDP is solved to obtain a certified upper bound, which is added to the dataset and used to guide the next iteration. Through this closed feedback loop, the agent actively selects both the geometric parameters and the polynomial structure of the certificate, enabling it to discover progressively tighter sphere-packing bounds.
• Figure 4: Scatter plot showing the degrees of polynomials discovered across all dimensions. Each point is a polynomial, with marker size proportional to its degree. We observe that the polynomials that appear most frequently, those with the highest counts (around 12), tend to have low degrees.
• Figure 5: Explored Regions of $(r,R)$ configurations in dimensions 14 (left) and 16 (right). Our system begins by broadly sampling candidate $(r,R)$ pairs (dark red points), before progressively shifting toward exploitation in regions predicted to yield improved upper bounds (light red region). The shaded red region denotes the neighbourhood of $(r,R)$ values where the tightest bounds have been achieved. Unlike prior mathematical approaches, which varied only a single parameter and therefore explored a restricted slice of the landscape, our method systematically searches across the space and uncovers configurations that were previously unexamined.