Refining Integration-by-Parts Reduction of Feynman Integrals - Using Machine Learning for Improved Heuristics

Aug 5, 2026

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Integration-by-parts reductions of Feynman integrals are a critical bottleneck in precision calculations for theoretical particle physics and gravitational-wave physics, where the efficiency strongly depends on heuristic choices for selecting integration-by-parts identities. This paper studies machine-learning methods to improve those heuristics. It applies funsearch, a genetic-programming variant leveraging code generation by a large language model, and strongly typed genetic programming to search and refine useful solutions. The methods reproduce state-of-the-art heuristics and identify a small improvement in at least one example.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages27

Reading Time1 h 8 min

AccessFree to download

Introduction

Integration-by-parts (IBP) identities and master integrals

Seeding strategies and heuristic impact

Background

Genetic algorithms and machine learning

Initial attempt: genetic algorithm

Exploration with Funsearch

Improved heuristics via strongly typed genetic programming

Conclusions and discussion

Appendix: IBP identities for the benchmark integral

University of Southern Denmark Refining Integration-by-Parts Reduction of Feynman Integrals with Machine Learning von Hippel, Matt; Wilhelm, Matthias Published in: Journal of High Energy Physics DOI: 10.1007/JHEP05(2025)185 Publication date: 2025 Document version: Final published version Document license: CC BY Citation for pulished version (APA): von Hippel, M. , & Wilhelm, M. (2025) . Refining Integration-by-Parts Reduction of Feynman Integrals with Machine Learning. Journal of High Energy Physics, 2025(5), Article 185. [https://doi.org/10.1007/JHEP05](https://doi.org/10.1007/JHEP05)(2025)185 Go to publication entry in University of Southern Denmark's Research Portal Terms of use This work is brought to you by the University of Southern Denmark. Unless otherwise specified it has been shared according to the terms for self-archiving. If no other license is stated, these terms apply: • You may download this work for personal use only. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying this open access version If you believe that this document breaches copyright please contact us providing details and we will investigate your claim. Please direct all enquiries to [puresupport@bib.sdu.dk](puresupport@bib.sdu.dk) Download date: 05. Aug. 2026 Published for SISSA by Springer Received: February 17, 2025 Accepted: April 14, 2025 Published: May 21, 2025 Refining Integration-by-Parts Reduction of Feynman Integrals with Machine Learning Matt von Hippel a and Matthias Wilhelm a,b a Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark b Center for Quantum Mathematics, Department of Mathematics and Computer Science, University of Southern Denmark, Campusvej 55, 5230 Odense M, Denmark E-mail: [mattvonhippel@gmail.com](mattvonhippel@gmail.com) , [mwilhelm@imada.sdu.dk](mwilhelm@imada.sdu.dk) Abstract: Integration-by-parts reductions of Feynman integrals pose a frequent bottleneck 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 integrationby-parts solvers, and in one example find a small advance on this state of the art. Keywords: Scattering Amplitudes, Automation, Electroweak Precision Physics ArXiv ePrint: 2502.05121 Open Access, © The Authors. Article funded by SCOAP3 . [https://doi.org/10.1007/JHEP05](https://doi.org/10.1007/JHEP05) (2025)185 J HEP05(2025)185 Contents 1 Introduction 1 2 Background 3 2.1 Integration-by-parts identities for Feynman integrals 3 2.2 Genetic algorithms and machine learning 7 3 Initial attempt: genetic algorithm 9 4 Exploration with Funsearch 10 5 Improved heuristics via strongly typed genetic programming 17 6 Conclusions and discussion 19 A Integration-by-parts identities for the benchmark integral 21 1 Introduction Perturbative Quantum Field Theory has proven to be a vastly successful theoretical framework for calculating precision predictions, with applications ranging from collider physics to gravitational-wave physics. A crucial step in the calculation of precision predictions is the reduction of the occurring Feynman integrals to a much smaller set of so-called master integrals, using integration-by-parts (IBP) identities [1–3] . This IBP reduction is a major b

FAQs

Why is integration-by-parts (IBP) reduction a bottleneck in precision calculations?

IBP reductions require choosing integration-by-parts identities and seeds whose heuristic quality strongly affects runtime. The computational cost can reach hundreds of thousands of CPU hours in current applications.

What machine-learning approaches are used to improve the IBP seeding/heuristics?

The paper uses funsearch (genetic programming with LLM-based code generation) and strongly typed genetic programming. It also includes versions of genetic algorithms as part of the search process.

Do the proposed methods reproduce existing best heuristics?

Yes. The approaches manage to rediscover state-of-the-art heuristics recently incorporated into IBP solvers, and in one example they find a small advance over that state of the art.

Aug 5, 2026

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