Flow-based Extremal Mathematical Structure Discovery

2026-01-25

AI Summary

This work addresses the challenge of efficiently discovering rare extremal geometric configurations in high-dimensional non-convex spaces by introducing FlowBoost, a novel framework that achieves the first closed-loop coupling between generative modeling and optimization objectives. FlowBoost integrates geometry-aware conditional flow matching, reward-guided policy optimization, and stochastic local search to directly optimize the generative process, ensuring geometric feasibility while enabling efficient reward backpropagation. Compared to conventional approaches and large language model (LLM) baselines, FlowBoost substantially reduces required training data, computational time, and iteration counts. It attains or surpasses state-of-the-art results on several benchmark problems—including sphere packing, circle packing, the Heilbronn triangle problem, and star discrepancy minimization—setting a new lower bound for the circle packing problem.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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.

Problem

• extremal structures
• geometric optimization
• nonconvex landscapes
• sphere packing
• Heilbronn triangle problem

Innovation

• flow matching
• closed-loop generation
• geometric optimization
• reward-guided policy optimization
• extremal structure discovery