Efficient AI-Inspired Reduction of Feynman Integrals via Tube Seeding

2026-06-09

AI Summary

This work addresses the computational bottleneck in integration-by-parts (IBP) reduction of high-loop Feynman integrals, where the number of seed integrals grows polynomially with the numerator rank. To overcome this limitation, the authors propose a machine-learning-inspired “tubular seed” strategy that restricts seed selection to a sparse, zigzag path within a narrow tubular region connecting the target integral to master integrals. This approach reduces the seed count to a linear dependence on the numerator rank. Combined with the Laporta algorithm, finite-field numerical momenta, and block-wise processing, the method substantially lowers computational complexity and memory usage. It enables efficient reduction of non-planar two-loop five-point integrals up to rank 20 and the complete set of top-level rank-10 integrals, thereby surpassing the scalability limits of conventional techniques and facilitating high-precision phenomenological calculations.

Abstract

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.

Problem

• Feynman integrals
• integration-by-parts reduction
• machine learning
• Laporta algorithm
• computational bottleneck

Innovation

• Feynman integrals
• integration-by-parts reduction
• machine learning
• sparse seeding
• tube seeding