# Efficient AI-Inspired Reduction of Feynman Integrals via Tube Seeding

arXiv: https://arxiv.org/abs/2606.10698 · HTML: https://mlsanity.com/paper/2606.10698/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 90 references that are in the map.
- First posted on arXiv 9 Jun 2026.

## Builds on (most cited first, 13 of 13)

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- [Learning the Simplicity of Scattering Amplitudes](https://mlsanity.com/paper/2408.04720/) (2024)
- [Refining Integration-by-Parts Reduction of Feynman Integrals with Machine Learning](https://mlsanity.com/paper/2502.05121/) (2025)
- [Learning to Unscramble: Simplifying Symbolic Expressions via Self-Supervised Oracle Trajectories](https://mlsanity.com/paper/2603.11164/) (2026)
- [Learning to Unscramble Feynman Loop Integrals with SAILIR](https://mlsanity.com/paper/2604.05034/) (2026)
- [Uncovering Singularities in Feynman Integrals via Machine Learning](https://mlsanity.com/paper/2510.10099/) (2025)
- [Learning the S-matrix from data: Rediscovering gravity from gauge theory via symbolic regression](https://mlsanity.com/paper/2602.15169/) (2026)

## Abstract (arXiv)

In this paper, we use machine learning to discover a new seeding strategy for integration-by-parts reduction of Feynman integrals, which is a frequent bottleneck in state-of-the-art calculations in theoretical particle and gravitational-wave physics. Our strategy allows us to reduce multi-loop integrals with large numerator powers via essentially the standard Laporta algorithm but with a sparse selection of seed integrals that grows only linearly with the numerator power, whereas existing strategies lead to growth with a polynomial power that increases with the complexity of the integral being reduced. The seeds are restricted to a thin tube-like region that connects the target integral to the master integrals along a zigzag path. We demonstrate the power of our approach by reducing non-planar 2-loop 5-point integrals of rank 20 with numerical kinematics over a finite field, which is prohibitively difficult for the Laporta algorithm with conventional seeding. Going beyond individual integrals, we further demonstrate the reduction of a complete set of top-level rank-10 integrals by dividing the target integrals into several chunks, each of which can be solved by our sparse seeding strategy with considerably less time and a significantly lower memory footprint than other state-of-the-art strategies, making the approach well-suited for phenomenological applications. We provide a proof-of-principle implementation on GitHub at https://github.com/andreslunagodoy/tube_seeding.