Flow-based Extremal Mathematical Structure Discovery

TL;DR

FlowBoost reframes mathematical structure discovery as a closed-loop SBO problem over continuous configuration spaces. It combines Conditional Flow Matching to model a transport to high-quality configurations, geometry-aware sampling to enforce hard geometric feasibility, and reward-guided fine-tuning with a teacher–student consistency term to push sampling toward extremal solutions while preserving diversity. Across sphere packing, circle packing, Heilbronn, and star discrepancy tasks, FlowBoost achieves competitive or superior results with orders of magnitude less outer-loop iterations and without reliance on Large Language Models. This approach unifies flow-based generative modeling, RL-inspired feedback, and geometry-aware constraints, offering a scalable tool for exploring extremal configurations in combinatorial geometry and beyond.

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.

Figures (42)

• Figure 1: Simulation-based optimization with closed-loop updates and action exploration. Gray dashed arrows indicate the open-loop baseline: generate candidates, select, and retrain on the selected set over many cycles. Green arrows indicate our closed-loop pipeline: samples produced by geometry-aware sampling are perturbed by an action exploration operator $\mathcal{E}(x'|x)$, scored by rewards $R(x)$, and used for online reward-guided fine-tuning of the flow model. The reward-weighted objective upweights high-reward samples via $w(R(x))$, while the teacher, student consistency term $\|v_\theta - v_{\mathrm{ref}}\|^2$ limits distribution shift and mitigates generative collapse.
• Figure 2: Comparison of the minimum-excess metric for two local search heuristics (common circle counts). The physics-push heuristic yields consistently higher minimum-excess values than SRP/SRS, indicating worse configurations across the tested regime.
• Figure 3: Sphere packing in $d=3$. (a–c) Normalized histograms of minimum pairwise distance (log scale). In all cases, RG-CFM with final push (blue) recovers the training distribution and yields configurations exceeding the training maximum: $d_{\min} = 0.261231$ vs. $0.261027$ for $N=55$; $0.232539$ vs. $0.232529$ for $N=83$; $0.180671$ vs. $0.180350$ for $N=191$. (d) Ablation comparing vanilla CFM (orange) against reward-guided CFM (green) for $N=83$: reward guidance with action exploration shifts the distribution toward higher-quality samples ($d_{\min}^{\max} = 0.2236$ vs. $0.2157$), demonstrating that direct objective feedback improves generation quality prior to local refinement.
• Figure 4: 3D Sphere Packing in Cube, $n=89$. The comparison between PatternBoost, RG-CFM, and their pushed version with the training data. Our Reward-guided fine tuning alone could provide the same performance as the expensive PatternBoost and SRP Push combined, while the pushed version of RG-CFM could improve the max over the training data.
• Figure 5: Sphere packing in $d=12$, $N=31$. Normalized histograms of minimum pairwise distance (log scale).