[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-127245-en":3,"doc-seo-127245-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},127245,2336475104362,"Eden","https://ap-avatar.wpscdn.com/avatar/22000c4c46a41b752dd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786595829695023868",8,"Research & Report","Refining Integration-by-Parts Reduction of Feynman Integrals - Using Machine Learning for Improved Heuristics","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.","University of Southern Denmark  \nRefining Integration-by-Parts Reduction of Feynman Integrals with Machine Learning  \nvon Hippel, Matt; Wilhelm, Matthias  \nPublished in:  \nJournal of High Energy Physics  \nDOI:  \n10.1007/JHEP05(2025)185  \nPublication date: 2025  \nDocument version:  \nFinal published version  \nDocument license: CC BY  \nCitation for pulished version (APA):  \nvon 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.  \n[https://doi.org/10.1007/JHEP05](https://doi.org/10.1007/JHEP05)(2025)185  \nGo to publication entry in University of Southern Denmark's Research Portal  \nTerms of use  \nThis work is brought to you by the University of Southern Denmark.  \nUnless otherwise specified it has been shared according to the terms for self-archiving.  \nIf no other license is stated, these terms apply:  \n• You may download this work for personal use only.  \n• You may not further distribute the material or use it for any profit-making activity or commercial gain  \n• You may freely distribute the URL identifying this open access version  \nIf 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)  \nDownload date: 05. Aug. 2026  \nPublished for SISSA by  Springer  \nReceived: February 17, 2025  \nAccepted: April 14, 2025  \nPublished: May 21, 2025  \nRefining Integration-by-Parts Reduction of Feynman Integrals with Machine Learning  \nMatt von Hippel a and Matthias Wilhelm a,b  \na Niels Bohr International Academy, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark  \nb Center for Quantum Mathematics, Department of Mathematics and Computer Science, University of Southern Denmark,  \nCampusvej 55, 5230 Odense M, Denmark  \nE-mail: [mattvonhippel@gmail.com](mattvonhippel@gmail.com) , [mwilhelm@imada.sdu.dk](mwilhelm@imada.sdu.dk)  \nAbstract: 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.  \nKeywords: Scattering Amplitudes, Automation, Electroweak Precision Physics  \nArXiv ePrint: 2502.05121  \nOpen Access, © The Authors.  \nArticle funded by SCOAP3 . [https://doi.org/10.1007/JHEP05](https://doi.org/10.1007/JHEP05) (2025)185  \nJ HEP05(2025)185  \nContents  \n1 Introduction 1  \n2 Background 3  \n2.1 Integration-by-parts identities for Feynman integrals 3  \n2.2 Genetic algorithms and machine learning 7  \n3 Initial attempt: genetic algorithm 9  \n4 Exploration with Funsearch 10  \n5 Improved heuristics via strongly typed genetic programming 17  \n6 Conclusions and discussion 19  \nA Integration-by-parts identities for the benchmark integral 21  \n1 Introduction  \nPerturbative 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","cbCaihN2kbwYAhvF","https://ap.wps.com/l/cbCaihN2kbwYAhvF","pdf",477814,1,27,"English","en",105,"# Introduction\n## Integration-by-parts (IBP) identities and master integrals\n## Seeding strategies and heuristic impact\n# Background\n## Genetic algorithms and machine learning\n# Initial attempt: genetic algorithm\n# Exploration with Funsearch\n# Improved heuristics via strongly typed genetic programming\n# Conclusions and discussion\n## Appendix: IBP identities for the benchmark integral","[{\"question\":\"Why is integration-by-parts (IBP) reduction a bottleneck in precision calculations?\",\"answer\":\"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.\"},{\"question\":\"What machine-learning approaches are used to improve the IBP seeding/heuristics?\",\"answer\":\"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.\"},{\"question\":\"Do the proposed methods reproduce existing best heuristics?\",\"answer\":\"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.\"}]","Refining Integration-by-Parts Reduction of Feynman Integrals - Using Machine Learning for Improved Heuristics | PDF",1785937720,68,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"refining-integration-by-parts-reduction-of-feynman-integrals-using-machine-learning-for-improved-heuristics","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/refining-integration-by-parts-reduction-of-feynman-integrals-using-machine-learning-for-improved-heuristics/127245/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-22","2026-08-05",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why is integration-by-parts (IBP) reduction a bottleneck in precision calculations?","Question",{"text":76,"@type":77},"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.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What machine-learning approaches are used to improve the IBP seeding/heuristics?",{"text":81,"@type":77},"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.",{"name":83,"@type":74,"acceptedAnswer":84},"Do the proposed methods reproduce existing best heuristics?",{"text":85,"@type":77},"Yes. 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