# Learning to Unscramble Feynman Loop Integrals with SAILIR

arXiv: https://arxiv.org/abs/2604.05034 · HTML: https://mlsanity.com/paper/2604.05034/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 37 references that are in the map.
- First posted on arXiv 6 Apr 2026.

## Builds on (most cited first, 4 of 4)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Passage Re-ranking with BERT](https://mlsanity.com/paper/1901.04085/) (2019)
- [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)

## Built on by (most cited first, 3 of 3)

- [When Does Critique Improve AI-Assisted Theoretical Physics? SCALAR: Structured Critic--Actor Loop for Agentic Reasoning](https://mlsanity.com/paper/2605.06772/) (2026)
- [Efficient AI-Inspired Reduction of Feynman Integrals via Tube Seeding](https://mlsanity.com/paper/2606.10698/) (2026)
- [Machine learning for the LHC physics program: a 2025-2026 stocktake](https://mlsanity.com/paper/2609.32874/) (2026)

## Abstract (arXiv)

Integration-by-parts (IBP) reduction of Feynman integrals to master integrals is a key computational bottleneck in precision calculations in high-energy physics. Traditional approaches based on the Laporta algorithm require solving large systems of equations, leading to memory consumption that grows rapidly with integral complexity. We present SAILIR (Self-supervised AI for Loop Integral Reduction), a new machine learning approach in which a transformer-based classifier guides the reduction of integrals one step at a time in a fully online fashion. The classifier is trained in an entirely self-supervised manner on synthetic data generated by a scramble/unscramble procedure: known reduction identities are applied in reverse to build expressions of increasing complexity, and the classifier learns to undo these steps. When combined with beam search and a highly parallelized, asynchronous, single-episode reduction strategy, SAILIR can reduce integrals of arbitrarily high weight with bounded memory. We benchmark SAILIR on the two-loop triangle-box topology, comparing against the state-of-the-art IBP reduction code Kira across 16 integrals of varying complexity. While SAILIR is slower in wall-clock time, its per-worker memory consumption remains approximately flat regardless of integral complexity, in contrast to Kira whose memory grows rapidly with complexity. For the most complex integrals considered here, SAILIR uses only 40% of the memory of Kira while achieving comparable reduction times. This demonstrates a fundamentally new paradigm for IBP reduction in which the memory bottleneck of Laporta-based approaches could be entirely overcome, potentially opening the door to precision calculations that are currently intractable.