Refining Integration-by-Parts Reduction of Feynman Integrals with Machine Learning

TL;DR

The paper tackles the IBP reduction bottleneck by searching for improved seed-seeding heuristics with machine learning. It combines funsearch, a text-based genetic-programming approach guided by a Large Language Model, and strongly typed genetic programming to rediscover and refine seed strategies for IBP systems. On the two-loop triangle-box benchmark, both methods reproduce the state-of-the-art seeding heuristics and, in one instance, achieve a notable reduction in seed count from tens of thousands to as few as 88, illustrating a practical uplift in reduction efficiency. The work demonstrates interpretable, data-driven heuristics that could be integrated into IBP software and scaled to more complex integrals, with potential workflow optimizations via dynamic seeding during rational reconstruction.

Abstract

Integration-by-parts reductions of Feynman integrals pose a frequent bottle-neck in state-of-the-art calculations in theoretical particle and gravitational-wave physics, and rely on heuristic approaches for selecting integration-by-parts identities, whose quality heavily influences the performance. In this paper, we investigate the use of machine-learning techniques to find improved heuristics. We use funsearch, a genetic programming variant based on code generation by a Large Language Model, in order to explore possible approaches, then use strongly typed genetic programming to zero in on useful solutions. Both approaches manage to re-discover the state-of-the-art heuristics recently incorporated into integration-by-parts solvers, and in one example find a small advance on this state of the art.

Figures (9)

• Figure 1: The two-loop triangle-box integral we used to benchmark different heuristics. The quantities at the arrows specify the momenta flowing through the edges of the graph.
• Figure 2: A tree diagram for a simple program given in the main text.
• Figure 3: Priority function used for the initial prompt to funsearch, corresponding to a golden rule system with $d_\textrm{max}=1$.
• Figure 4: A function generated by funsearch which gives 214 seeds for our test case.
• Figure 5: A function generated by funsearch which gives 92 seeds for our test case, equivalent to improved seeding with $d_\textrm{max}=0$ and $l=4$.