[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-125194-en":3,"doc-seo-125194-105":30,"detail-sidebar-cat-0-en-105":91},{"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},125194,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Dirac-equation signal processing - Physics boosts topological machine learning","Topological signals are features defined on both nodes and edges of networks, and their signal processing is central to topological machine learning. Prior methods often separate node and edge signals and assume smoothness or harmonicity with respect to the higher-order Laplacian, which can fail in real data. A Dirac-equation signal processing framework reconstructs node-and-edge signals jointly using spectral properties of the topological Dirac operator. Relativistic dispersion relations guide quality assessment, and experiments show improved performance even for non-harmonic and nontrivial linear-combination signals.","PNAS Nexus , 2025, 4 , pgaf139  \n[https://doi.org/10.1093/pnasnexus/pgaf139](https://doi.org/10.1093/pnasnexus/pgaf139)[ ](https://doi.org/10.1093/pnasnexus/pgaf139)Advance access publication 2 May 2025 Research Report  \nDirac-equation signal processing: Physics boosts topological machine learning  \nRunyue Wanga , Yu Tian b , c , Pietro Liòd and Ginestra Bianconi a ,*  \naCentre for Complex Systems, School of Mathematical Sciences, Queen Mary University of London, London E1 4NS, United Kingdom bNordita, KTH Royal Institute of Technology and Stockholm University, Stockholm SE-106 91, Sweden  \ncCenter for Systems Biology Dresden, 108 Pfotenhauerstraße, Dresden 01307, Germany  \ndDepartment of Computer Science and Technology, University of Cambridge, Cambridge CB3 0FA, United Kingdom  \n*To whom correspondence should be addressed: Email: [g.bianconi@qmul.ac.uk](g.bianconi@qmul.ac.uk)[ ](g.bianconi@qmul.ac.uk)Edited By Attila Szolnoki  \nAbstract  \nTopological signals are variables or features associated with both nodes and edges of a network. Recently, in the context of topological machine learning, great attention has been devoted to signal processing of such topological signals. Most of the previous topological signal processing algorithms treat node and edge signals separately and work under the hypothesis that the true signal is smooth and/or well approximated by a harmonic eigenvector of the higher-order Laplacian, which may be violated in practice. Here, we propose Dirac-equation signal processing, a framework for efficiently reconstructing true signals on nodes and edges, also if they are not smooth or harmonic, by processing them jointly. The proposed physics-inspired algorithm is based on the spectral properties of the topological Dirac operator. It leverages the mathematical structure of the topological Dirac equation to boost the performance of the signal processing algorithm. We discuss how the relativistic dispersion relation obeyed by the topological Dirac equation can be used to assess the quality of the signal reconstruction. Finally, we demonstrate the improved performance of the algorithm with respect to previous algorithms. Specifically, we show that Dirac-equation signal processing can also be used efficiently if the true signal is a nontrivial linear combination of more than one eigenstate of the Dirac equation, as it generally occurs for real signals.  \nKeywords: topological signals, topological machine learning, topological signal processing, topological Dirac equation, networks  \nSignificance Statement  \nMachine learning and physics have a long-standing relation. Most notably, neural networks are based on statistical mechanics’ early breakthroughs in understanding learning, such as the Hopfield model and the Boltzmann machines. Here, we show that theoretical physics insights can also boost topological signal processing. Topological signal processing of node and edge signals defined on networks is gaining large attention, but the node and edge signals are usually treated separately. The topological Dirac equation generalizes the Kogut–Susskind staggered fermions, and can be used to jointly process node and edge signals when adopted to regularize the signal processing loss function. Here, we demonstrate that the proposed Dirac-equation signal processing boosts the performance of topological signal processing when the true signal is not harmonic.  \nIntroduction  \nPhysics and AI are strongly related (1) as the theory of information is at the core of natural physical systems as well as of learning. Indeed, it is not by chance that the theory of learning has its roots in physically inspired models such as the Hopfield model (2) strongly related to statistical mechanics of disordered systems (3 , 4) . In more recent developments of the field, however, not only classical statistical mechanics has become relevant to understanding learning but also high-energy physics (5 , 6), quantum physics (7), and network science (8","cbCaicWpV8rJIMpR","https://ap.wps.com/l/cbCaicWpV8rJIMpR","pdf",873093,1,12,"English","en",105,"# Abstract\n# Significance Statement\n# Introduction","[{\"question\":\"What problem does the proposed Dirac-equation signal processing address?\",\"answer\":\"It targets efficient reconstruction of true signals on both nodes and edges when node/edge signals cannot be assumed smooth or harmonic. The method processes them jointly rather than separately.\"},{\"question\":\"How is the topological Dirac equation used in the algorithm?\",\"answer\":\"The approach leverages spectral properties of the topological Dirac operator and uses the structure of the topological Dirac equation to boost signal reconstruction performance.\"},{\"question\":\"How is the reconstruction quality assessed?\",\"answer\":\"The algorithm uses the relativistic dispersion relation obeyed by the topological Dirac equation to evaluate the quality of the reconstructed signal.\"}]","Dirac-equation signal processing - Physics boosts topological machine learning | PDF",1785897303,30,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"dirac-equation-signal-processing-physics-boosts-topological-machine-learning","",{"@graph":36,"@context":85},[37,54,68],{"@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/dirac-equation-signal-processing-physics-boosts-topological-machine-learning/125194/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-05",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the proposed Dirac-equation signal processing address?","Question",{"text":75,"@type":76},"It targets efficient reconstruction of true signals on both nodes and edges when node/edge signals cannot be assumed smooth or harmonic. The method processes them jointly rather than separately.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How is the topological Dirac equation used in the algorithm?",{"text":80,"@type":76},"The approach leverages spectral properties of the topological Dirac operator and uses the structure of the topological Dirac equation to boost signal reconstruction performance.",{"name":82,"@type":73,"acceptedAnswer":83},"How is the reconstruction quality assessed?",{"text":84,"@type":76},"The algorithm uses the relativistic dispersion relation obeyed by the topological Dirac equation to evaluate the quality of the reconstructed signal.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,122,127,130,134],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":29,"slug":121},"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]