Flow-based Extremal Mathematical Structure Discovery

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 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.

• The paper presents FlowBoost, a closed-loop framework that uses geometry-aware conditional flow matching and reward-guided policy optimization to efficiently discover extremal structures in high-dimensional spaces.
• The paper demonstrates FlowBoost's superior performance on tasks like sphere packing, Heilbronn triangle, circle packing, and star discrepancy minimization, achieving high quality with fewer iterations.
• The paper reveals that closed-loop updates and explicit action exploration significantly enhance sample quality and convergence, overcoming limitations of traditional open-loop methods.

FlowBoost: A Closed-Loop Flow-Based Framework for Extremal Mathematical Structure Discovery

Introduction

This work presents FlowBoost, a closed-loop generative modeling framework for the discovery of extremal mathematical structures in high-dimensional, geometrically constrained settings. The methodology fundamentally targets scenarios where traditional symbolic and discrete AI-driven approaches are inadequate, specifically continuous optimization tasks characterized by complex constraint manifolds and rugged, multimodal objective functions. FlowBoost integrates three core components: (1) a geometry-aware conditional flow matching (CFM) model for continuous configuration generation, (2) reward-guided policy optimization with explicit exploration, and (3) stochastic local search (SRP) for both data curation and sample refinement.

Limitations of Prior Approaches

Prior generative discovery pipelines—including PatternBoost (Charton et al., 2024) and LLM-driven evolutionary agents such as AlphaEvolve (Novikov et al., 16 Jun 2025)—face critical limitations in continuous geometric optimization. Their primary drawbacks are the necessity for discretization, reliance on tokenized or programmatic representations, computational dependence on large unfrozen LLMs, and the lack of direct closed-loop objective feedback. These methods typically deploy open-loop iteration, learning to mimic elite samples without targeted gradient-based improvement, thereby requiring extensive retraining cycles and offering no theoretical guarantees of convergence toward near-optimal solutions.

FlowBoost Architecture

Conditional Flow Matching (CFM) and Geometry-Aware Sampling

The backbone of FlowBoost is a CFM generative model that parameterizes a velocity field—via a permutation-equivariant neural architecture—mapping a prior distribution to the manifold of high-quality, feasible configurations in the ambient Euclidean space. The model is trained on empirical data refined by local search and regularized with geometric penalty terms (e.g., soft-overlap energies) to enforce approximate feasibility, moving beyond simple post-hoc rejection or repair. During sampling, FlowBoost executes geometry-aware integration: ODE steps are interleaved with constraint projections (Gauss-Newton updates, proximal relaxations) that ensure strict satisfaction of feasibility constraints throughout the generative process.

Figure 1: The minimum-excess metric for local search heuristics reveals that SRP/SRS is consistently superior to simple physics-push baselines.

Closed-Loop Reward-Guided Optimization and Action Exploration

To transition from passive distribution matching toward systematic optimization, FlowBoost introduces reward-guided CFM (RG-CFM). Here the model parameters are fine-tuned by importance-weighted objectives, where reward weights are dynamically computed via $z$-scored exponentiation of the target metric (e.g., minimum distance, area, or sum of radii), promoting upweighting of rare, high-function-value samples. Consistency regularization, inspired by self-supervised learning, constrains the student policy to remain close to a pretrained teacher baseline, preventing collapse to degenerate deterministic policies and ensuring maintenance of sample diversity. FlowBoost further augments policy updates with explicit action exploration operators that perform smooth, constraint-informed perturbations of generated samples, expanding support and enhancing OOD discovery.

Experimental Evaluation

Sphere Packing in Hypercubes ($d=3,12$)

FlowBoost is evaluated on $3$-dimensional and $12$-dimensional sphere packing in cubes, maximizing the effective sphere radius subject to non-overlap and containment. Compared to PatternBoost and SRP-only baselines, RG-CFM achieves comparable or improved best minimum separations in drastically fewer iterations and sample counts, with high computational efficiency (single-GPU scale). Notably, in high dimensions where local search stagnates, RG-CFM attains configurations that exceed the maximum of extensive SRP reference sets after only one or two fine-tuning rounds.

Figure 2: For $N=55$, RG-CFM (pushed) distribution demonstrates a shift toward larger minimum separations over SRP-generated training data.

Figure 3: For $N=89$, RG-CFM matches PatternBoost+SRP performance and its pushed variant achieves better-than-training data results.

Figure 4: In $d=12$, $N=31$, generated and pushed samples quickly match and slightly surpass the training set quality.

The pipeline exhibits significant right-shifting and narrowing of best-objective-value histograms with successive closed-loop updates, indicating that mass is efficiently allocated to high-quality basins. The average quality of generated samples increases monotonically with boosting rounds, even when the global optimum is nearly saturated.

Figure 5: Large training sets and long per-iteration model training lead to pronounced right-shifts in the $d_{\min}$ distribution for multiple $N$.

Figure 6: For $N=191$, even with many short iterations, FlowBoost consistently improves average sample quality over $100+$ iterations.

Heilbronn Triangle Problem

For the classical Heilbronn triangle problem, FlowBoost refines $n$-point configurations for maximal minimal triangle area. The smooth, local-to-global FlowBoost pipeline reliably repairs geometric degeneracies in generated samples and, with rewards and iteration, matches or slightly exceeds the elite set, surpassing the best-known results from SRP or L-BFGS-only approaches.

Figure 7: Distribution of $A_{\min}$ for $n=13$ after training, generation, and refinement steps, with boosting rounds clearly right-shifting the tail.

Figure 8: Same as above for $n=15$; closed-loop iteration increases the occurrence of high $A_{\min}$ configurations.

Figure 9: Geometric visualization of the best $n=13$ and $n=15$ configurations found by FlowBoost, with minimum-area triangles highlighted.

Circle Packing: Sum of Radii Maximization

FlowBoost is adapted for unit-square circle packings, targeting maximal sum of radii for fixed $N$. The generative model proposes centers, radii are solved by LP, and final refinement is performed with SRP and L-BFGS. Strong numerical outcomes are reported: for $n=26$ and $n=32$, FlowBoost achieves sums that strictly exceed AlphaEvolve’s best-reported lower bounds, producing these cases with higher frequency and efficiency.

Figure 10: $n=26$, distribution evolution after two boosting rounds shows consistent improvement in right tail ($\sum_i r_i$).

Figure 11: $n=30$, stable improvement in the extreme right; best values maintained by FlowBoost.

Figure 12: $n=32$, sustained right-shifting in $\sum_i r_i$ over three iterations; superior tail density.

Figure 13: Visualization of best $n=26, 30, 32$ circle packings after FlowBoost, all at/beyond state-of-the-art in $\sum r_i$.

Star Discrepancy Minimization

For star discrepancy minimization in $[0,1]^2$, FlowBoost's local search and boosting pipeline are leveraged to discover point sets with minimal worst-case anchored box error. The final pushed samples not only match but, in several cases, improve on existing open benchmarks.

Figure 14: For $N=20$, the pushed distribution after boosting achieves best values beyond the training maximum.

Figure 15: For $N=60$, raw generative samples lag, but the final push with local search reliably recovers state-of-the-art discrepancy.

Figure 16: Visualizations of best low-discrepancy point sets returned by FlowBoost for $N=20, 60$ surpass existing constructions.

Computational Efficiency and Theoretical Implications

FlowBoost achieves state-of-the-art or superlative results in a fraction of the compute and sample regimes dictated by open-loop or LLM-dependent baselines. The compact parameterization (2M vs. 10B+ for LLMs), model reuse across problem domains, and lack of tokenization overhead enables individual researchers to perform competitive discovery on commodity hardware.

Theoretically, FlowBoost offers a systematic mechanism for direct search distribution improvement, resolving the inefficiency and stochasticity that characterizes open-loop density-matching frameworks. The closed-loop reward-guided updates, combined with geometric inductive biases encoded into the generator and explicit support expansion via action exploration, strongly mitigate mode collapse and support optimal convergence properties in constrained continuous combinatorial regimes. FlowBoost’s paradigm—de novo mathematical structure design—effectively bridges combinatorial, geometric, and simulation-based optimization methodologies.

Conclusion

FlowBoost substantively advances the closed-loop generative modeling paradigm for extremal mathematical structure discovery, demonstrating robust empirical improvement over state-of-the-art methods across a suite of geometric optimization benchmarks. The synergy of conditional flow-based modeling, domain-informed sampling, and reward-guided learning produces a framework that is sample-efficient, compute-efficient, and theoretically principled for constrained continuous optimization problems in mathematics.

Further work may extend the FlowBoost blueprint to higher-dimensional discrepancy problems, algebraic energy minimization, general constraint satisfaction, and other simulation-based continuous design problems, with potential for automated conjecture generation and experimental mathematics at scale.