{"id":"https://openalex.org/W7136233606","doi":"https://doi.org/10.48550/arxiv.2603.12982","title":"RUNNs: Ritz-Uzawa Neural Networks for Solving Variational Problems","display_name":"RUNNs: Ritz-Uzawa Neural Networks for Solving Variational Problems","publication_year":2026,"publication_date":"2026-03-13","ids":{"openalex":"https://openalex.org/W7136233606","doi":"https://doi.org/10.48550/arxiv.2603.12982"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2603.12982","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.12982","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2603.12982","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5129616861","display_name":"Pablo Herrera","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Herrera, Pablo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065057468","display_name":"Jamie M. Taylor","orcid":"https://orcid.org/0000-0002-5423-828X"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Taylor, Jamie M.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5129635618","display_name":"Carlos Uriarte","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Uriarte, Carlos","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5072003067","display_name":"Ignacio Muga","orcid":"https://orcid.org/0000-0003-4430-5167"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Muga, Ignacio","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5082696717","display_name":"David Pardo","orcid":"https://orcid.org/0000-0002-1101-2248"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Pardo, David","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5007817923","display_name":"Kristoffer G. van der Zee","orcid":"https://orcid.org/0000-0002-6830-8031"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"van der Zee, Kristoffer G.","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":6,"corresponding_author_ids":["https://openalex.org/A5129616861"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.973800003528595,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.973800003528595,"subfield":{"id":"https://openalex.org/subfields/3109","display_name":"Statistical and Nonlinear Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.011099999770522118,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.0019000000320374966,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/initialization","display_name":"Initialization","score":0.7156999707221985},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.6062999963760376},{"id":"https://openalex.org/keywords/robustness","display_name":"Robustness (evolution)","score":0.5654000043869019},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.4551999866962433},{"id":"https://openalex.org/keywords/spectral-density","display_name":"Spectral density","score":0.4212999939918518},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.400299996137619},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.37869998812675476},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.3707999885082245}],"concepts":[{"id":"https://openalex.org/C114466953","wikidata":"https://www.wikidata.org/wiki/Q6034165","display_name":"Initialization","level":2,"score":0.7156999707221985},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6062999963760376},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.5654000043869019},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5281000137329102},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5095000267028809},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.4551999866962433},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4307999908924103},{"id":"https://openalex.org/C168110828","wikidata":"https://www.wikidata.org/wiki/Q1331626","display_name":"Spectral density","level":2,"score":0.4212999939918518},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.400299996137619},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.39340001344680786},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.37869998812675476},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.3707999885082245},{"id":"https://openalex.org/C176321772","wikidata":"https://www.wikidata.org/wiki/Q1430640","display_name":"Numerical stability","level":3,"score":0.35589998960494995},{"id":"https://openalex.org/C2776257435","wikidata":"https://www.wikidata.org/wiki/Q1576430","display_name":"Bandwidth (computing)","level":2,"score":0.35179999470710754},{"id":"https://openalex.org/C196083921","wikidata":"https://www.wikidata.org/wiki/Q7915758","display_name":"Variance (accounting)","level":2,"score":0.34869998693466187},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.33239999413490295},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.3305000066757202},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.3167000114917755},{"id":"https://openalex.org/C30049272","wikidata":"https://www.wikidata.org/wiki/Q6555326","display_name":"Spectral density estimation","level":3,"score":0.313400000333786},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.2678999900817871},{"id":"https://openalex.org/C86941820","wikidata":"https://www.wikidata.org/wiki/Q6865391","display_name":"Minimisation (clinical trials)","level":2,"score":0.2669999897480011},{"id":"https://openalex.org/C2778112365","wikidata":"https://www.wikidata.org/wiki/Q3511065","display_name":"Sequence (biology)","level":2,"score":0.26159998774528503},{"id":"https://openalex.org/C91575142","wikidata":"https://www.wikidata.org/wiki/Q1971426","display_name":"Optimal control","level":2,"score":0.26089999079704285},{"id":"https://openalex.org/C137836250","wikidata":"https://www.wikidata.org/wiki/Q984063","display_name":"Optimization problem","level":2,"score":0.2587999999523163}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2603.12982","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.12982","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2603.12982","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2603.12982","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[{"display_name":"No poverty","id":"https://metadata.un.org/sdg/1","score":0.4422728717327118}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Solving":[0],"Partial":[1],"Differential":[2],"Equations":[3],"(PDEs)":[4],"using":[5],"neural":[6,25],"networks":[7],"presents":[8],"different":[9],"challenges,":[10],"including":[11],"integration":[12],"errors":[13],"and":[14,64,93,115,165,180],"spectral":[15,122],"bias,":[16],"often":[17,34],"leading":[18],"to":[19,60,161],"poor":[20],"approximations.":[21],"In":[22],"addition,":[23],"standard":[24,125,196],"network-based":[26],"methods,":[27],"such":[28],"as":[29,71],"Physics-Informed":[30],"Neural":[31,53],"Networks":[32,54],"(PINNs),":[33],"lack":[35],"stability":[36],"when":[37],"dealing":[38],"with":[39],"PDEs":[40],"characterized":[41],"by":[42],"low-regularity":[43],"solutions.":[44],"To":[45],"address":[46,120],"these":[47],"limitations,":[48],"we":[49,119],"introduce":[50],"the":[51,69,100,121,142,155,171],"Ritz--Uzawa":[52],"(RUNNs)":[55],"framework,":[56],"an":[57,83],"iterative":[58,84],"methodology":[59],"solve":[61],"strong,":[62],"weak,":[63],"ultra-weak":[65,116],"variational":[66],"formulations.":[67],"Rewriting":[68],"PDE":[70],"a":[72,79,104,128,135,183,188,193],"sequence":[73],"of":[74,124,149,173],"Ritz-type":[75],"minimization":[76],"problems":[77],"within":[78],"Uzawa":[80],"loop":[81],"provides":[82],"framework":[85],"that,":[86],"in":[87,113],"specific":[88],"cases,":[89],"reduces":[90],"both":[91],"bias":[92,123],"variance":[94,106,110],"during":[95],"training.":[96],"We":[97],"demonstrate":[98,170],"that":[99],"strong":[101],"formulation":[102],"offers":[103],"passive":[105],"reduction":[107],"mechanism,":[108],"whereas":[109],"remains":[111],"persistent":[112],"weak":[114],"regimes.":[117],"Furthermore,":[118],"architectures":[126],"through":[127],"data-driven":[129],"frequency":[130],"tuning":[131],"strategy.":[132],"By":[133],"initializing":[134],"Sinusoidal":[136],"Fourier":[137],"Feature":[138],"Mapping":[139],"based":[140],"on":[141],"Normalized":[143],"Cumulative":[144],"Power":[145],"Spectral":[146],"Density":[147],"(NCPSD)":[148],"previous":[150],"residuals":[151],"or":[152],"their":[153],"proxies,":[154],"network":[156],"dynamically":[157],"adapts":[158],"its":[159],"bandwidth":[160],"capture":[162],"high-frequency":[163],"components":[164],"severe":[166],"singularities.":[167],"Numerical":[168],"experiments":[169],"robustness":[172],"RUNNs,":[174],"accurately":[175],"resolving":[176],"highly":[177],"oscillatory":[178],"solutions":[179],"successfully":[181],"recovering":[182],"discontinuous":[184],"$L^2$":[185],"solution":[186],"from":[187],"distributional":[189],"$H^{-2}$":[190],"source":[191],"--":[192],"scenario":[194],"where":[195],"energy-based":[197],"methods":[198],"fail.":[199]},"counts_by_year":[],"updated_date":"2026-05-05T08:41:31.759640","created_date":"2026-03-17T00:00:00"}