基于流方法的极值数学结构发现

Flow-based Extremal Mathematical Structure Discovery

January 25, 2026

作者: Gergely Bérczi, Baran Hashemi, Jonas Klüver

cs.AI

摘要

数学中极值结构的发现需要探索广阔且非凸的复杂空间，传统解析方法难以提供有效指导，而暴力搜索又往往不可行。我们提出FlowBoost——一种闭环生成式框架，通过融合三大组件来学习发现稀有极值几何结构：（一）几何感知的条件流匹配模型，可学习采样高质量构型；（二）结合行动探索的奖励引导策略优化，在保持多样性的同时直接优化生成过程以趋近目标；（三）用于训练数据生成与最终优化的随机局部搜索。相较于PatternBoost等基于过滤离散样本重训练的开放环路方法，或依赖冻结大语言模型作为进化变异算子的AlphaEvolve，FlowBoost在采样阶段强制保证几何可行性，并将奖励信号直接反馈至生成模型，形成闭环优化。该框架仅需少量训练样本和较短训练时长，就能将外层循环迭代次数降低数个量级，同时摆脱对大语言模型的依赖。我们在四个几何优化问题上验证了该框架：超立方体中的球体填充、半径和最大化的圆盘填充、Heilbronn三角形问题以及星形差异最小化。在多个案例中，FlowBoost发现的构型达到或超越了已知最优结果。针对圆盘填充问题，我们提升了已知下界的最佳记录，在显著减少计算资源消耗的同时超越了基于大语言模型的AlphaEvolve系统。

English

The discovery of extremal structures in mathematics requires navigating vast and nonconvex landscapes where analytical methods offer little guidance and brute-force search becomes intractable. We introduce FlowBoost, a closed-loop generative framework that learns to discover rare and extremal geometric structures by combining three components: (i) a geometry-aware conditional flow-matching model that learns to sample high-quality configurations, (ii) reward-guided policy optimization with action exploration that directly optimizes the generation process toward the objective while maintaining diversity, and (iii) stochastic local search for both training-data generation and final refinement. Unlike prior open-loop approaches, such as PatternBoost that retrains on filtered discrete samples, or AlphaEvolve which relies on frozen Large Language Models (LLMs) as evolutionary mutation operators, FlowBoost enforces geometric feasibility during sampling, and propagates reward signal directly into the generative model, closing the optimization loop and requiring much smaller training sets and shorter training times, and reducing the required outer-loop iterations by orders of magnitude, while eliminating dependence on LLMs. We demonstrate the framework on four geometric optimization problems: sphere packing in hypercubes, circle packing maximizing sum of radii, the Heilbronn triangle problem, and star discrepancy minimization. In several cases, FlowBoost discovers configurations that match or exceed the best known results. For circle packings, we improve the best known lower bounds, surpassing the LLM-based system AlphaEvolve while using substantially fewer computational resources.