Baran Hashemi

Flow-based Extremal Mathematical Structure Discovery 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 intr...

Flow-based extremal mathematical structure discovery. ~ Gergely Bérczi, Baran Hashemi, Jonas Klüver. arxiv.org/abs/2601.180... #AI4Math

1/ Are frontier LLMs the only path to AI math breakthroughs? I think not!
We introduce FlowBoost, A lightweight RL+Flow-Matching framework that discovers new extremal geometric structures, beating AlphaEvolve with 100–1000× less compute with zero-shot geometry-aware & reward-guided generation. 🚀

6/
📄 Paper: arxiv.org/pdf/2601.18005
💻 Code: github.com/berczig/Flow...
Built at
@au.dk
and
@maxplanck.de.
Happy to discuss methodology, results, or potential applications!
#AI4Mathematics #AI4Science #FlowMatching

5.2/ Key insight:
FlowBoost closes the loop: online reward-guided fine-tuning (with trust-region self-distillation) directly optimizes the flow policy — continual learning that converges in ~5 steps!

5.1/ Key insight:
Frame extremal math discovery as Simulation-Based Optimization (SBO) with a true closed feedback loop. Open-loop methods iterate blindly: generate → filter/select → retrain on elites. --> No direct signal pushes the policy toward rarer, higher-reward solutions.

4.2/ Results:
• Sphere packing 12d: Discover denser configurations than those produced by classical heuristics!
• Sphere packing 3d: Match or exceed the best previously reported packing fractions.

4.1/ Results:
• Circle packing: New records n=26 & 32, surpassing AlphaEvolve!
• Heilbronn Problem: Improve the minimum triangle area over the training dataset --> OOD sampling.

3.3/ Our key innovations:
• Trust-region Self-Distillation: Prevents collapse while enabling extrapolative OOD discovery.
Result: Convergence in ~10 updates vs 100–1000 in open-loop systems.

3.2/ Our key innovations:
• Closed-loop Reward-guided Fine-tuning: Online reward weighting + action exploration directly optimizes the flow toward rare high-score samples.

3.1/ Our key innovations:
• Geometry-Aware Sampling (GAS): Interleaves ODE integration with geometric constraint projections for feasible zero-shot samples on-the-fly.

2/ Extremal combinatorial geometry problems are rugged continuous landscapes, perfect for modern generative models with strong inductive biases. We ditch LLM code evolution for direct continuous generation + closed-loop optimization. We call the new paradigm, "de novo Mathematical Structure Design."

1/ Are frontier LLMs the only path to AI math breakthroughs? I think not!
We introduce FlowBoost, A lightweight RL+Flow-Matching framework that discovers new extremal geometric structures, beating AlphaEvolve with 100–1000× less compute with zero-shot geometry-aware & reward-guided generation. 🚀

Soon I will write a thread on our new work. Stay tuned!
#AI4Math

Tnx Kyle 🤜

We got accepted at #NeurIPS2025. I am very happy that I could merge my knowledge of Mathematics with AI to create sth new and useful for the community. ☺️

The paper: arxiv.org/abs/2505.17190
The code: github.com/Baran-phys/T...

Current AI research vibes:
- Let’s use LLM to do a baby science/math, after it doesn’t work, headline: LLM is bad at the baby math task —> guaranteed virality 😒
- Meanwhile, you develope a novel (non-LLM) method to solve this issue, report success on a deep math problem
—> naa, not enough drama🤦🏻

Another new result from the #NeurIPS rebuttal/discussion phase, our Tropical Transformer achieves much better length OOD performance across all algorithmic tasks, while being 3x-9x faster at inference and using 20% fewer parameters than the Universal Transformer (UT) models.

Tropical Attention: Neural Algorithmic Reasoning for Combinatorial Algorithms Dynamic programming (DP) algorithms for combinatorial optimization problems work with taking maximization, minimization, and classical addition in their recursion algorithms. The associated value functions correspond to convex polyhedra in the max plus semiring. Existing Neural Algorithmic Reasoning models, however, rely on softmax-normalized dot-product attention where the smooth exponential weighting blurs these sharp polyhedral structures and collapses when evaluated on out-of-distribution (OOD) settings. We introduce Tropical attention, a novel attention function that operates natively in the max-plus semiring of tropical geometry. We prove that Tropical attention can approximate tropical circuits of DP-type combinatorial algorithms. We then propose that using Tropical transformers enhances empirical OOD performance in both length generalization and value generalization, on algorithmic reasoning tasks, surpassing softmax baselines while remaining stable under adversarial attacks. We also present adversarial-attack generalization as a third axis for Neural Algorithmic Reasoning benchmarking. Our results demonstrate that Tropical attention restores the sharp, scale-invariant reasoning absent from softmax.

During #NeurIPS rebuttal, we have evaluated🌴Tropical Transformer on the Long Range Arena (LRA), achieving highly competitive results, placing 2nd🥈 overall in average accuracy.
Check out our paper: arxiv.org/abs/2505.17190
Our code: github.com/Baran-phys/T...

Cool. Will definitely do 👍

Interesting. I was not aware of aware if the challenges in the video subfield. But that makes sense given the context. We will definitely explore those benchmarks in the future. Thanks for the suggestions.

Tnx. We did not test yet on any other benshmarks. You mean algorithmic or language type benchmarks?

Interesting. I was not aware of this study. However, we did not just used tropical operations, we tried to simulate a concrete tropical circuit and do the message passing in the tropical space with the Generalized Hilbert metric as the kernel.

7/ Our message ✍️
Better reasoning might come not from bigger models, but from choosing the right algebra/geometry 🌴.
@petar-v.bsky.social @jalonso.bsky.social
#TropicalGeometry #NeuralAlgorithmicReasoning #AI4Math

6/ We also show that each Tropical attention head can function as a tropical gate in a tropical circuit, simulating any max-plus circuit.

5/ We benchmarked on 11 canonical combinatorial tasks. Tropical attention beat vanilla & adaptive softmax attention on all three OOD axes, Length, value and Adversarial attack generalization:

4/ Tropical Attention runs each head natively in max-plus. Result:
Strong OOD length generalization with sharp attention maps even in several algorithmic tasks, including the notorious Quickselect algorithm (Another settlement for the challenge identified by @mgalkin.bsky.social )

Image by Cowdery and Challas, featured in June 2009 Mathematics Magazine

3/ In the Tropical (max + ) geometry, “addition” is max, “multiplication” is +. Many algorithms already live here, carving exact polyhedral decision boundaries --> so why force them through exponential probabilities?
Let's ditch softmax, embrace the tropical semiring 🤯🍹.

Tropical Attention: Neural Algorithmic Reasoning for Combinatorial Algorithms Dynamic programming (DP) algorithms for combinatorial optimization problems work with taking maximization, minimization, and classical addition in their recursion algorithms. The associated value func...

2/ We introduce Tropical Attention -- the first Neural Algorithmic reasoner that operates in the Tropical semiring, achieving SOTA OOD performance on executing several combinatorial algorithms
arxiv.org/abs/2505.17190

🧵 Tropical Attention --> Softmax is out, Tropical max-plus is in 🦾
1/ 🔥Ever experinced softmax attention fade as sequences grow?
That blur is why many attention mechanisms stumble on algorithmic and reasoning tasks. Well, we have a Algebraic Geometric Tropical solution 🌴

I'm speaking about AI for enumerative geometry at the CMSA New Technologies in Mathematics seminar, on Wednesday.