Flow-based Extremal Mathematical Structure Discovery

TL;DR

FlowBoost is a novel generative framework that efficiently discovers extremal geometric structures by integrating geometry-aware sampling, reward-guided optimization, and local search, outperforming prior methods in accuracy and resource use.

Contribution

The paper introduces FlowBoost, a new closed-loop generative approach that enforces geometric feasibility and reduces training time compared to previous open-loop methods.

Findings

Successfully applied to four geometric problems with state-of-the-art results.

Improves lower bounds for circle packing, surpassing LLM-based methods.

Requires fewer resources and training iterations than prior approaches.

Abstract

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.