{"id":"https://openalex.org/W3004424267","doi":"https://doi.org/10.1137/19m1247620","title":"Layer-Parallel Training of Deep Residual Neural Networks","display_name":"Layer-Parallel Training of Deep Residual Neural Networks","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3004424267","doi":"https://doi.org/10.1137/19m1247620","mag":"3004424267"},"language":"en","primary_location":{"id":"doi:10.1137/19m1247620","is_oa":true,"landing_page_url":"https://doi.org/10.1137/19m1247620","pdf_url":null,"source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematics of Data Science","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://doi.org/10.1137/19m1247620","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5044666766","display_name":"Stefanie G\u00fcnther","orcid":"https://orcid.org/0000-0002-9263-6940"},"institutions":[{"id":"https://openalex.org/I153267046","display_name":"University of Kaiserslautern","ror":"https://ror.org/04zrf7b53","country_code":"DE","type":"education","lineage":["https://openalex.org/I153267046"]}],"countries":["DE"],"is_corresponding":true,"raw_author_name":"Stefanie G\u00fcnther","raw_affiliation_strings":["Univ. of Kaiserslautern (Germany)"],"affiliations":[{"raw_affiliation_string":"Univ. of Kaiserslautern (Germany)","institution_ids":["https://openalex.org/I153267046"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5076591073","display_name":"Lars Ruthotto","orcid":"https://orcid.org/0000-0003-0803-3299"},"institutions":[{"id":"https://openalex.org/I150468666","display_name":"Emory University","ror":"https://ror.org/03czfpz43","country_code":"US","type":"education","lineage":["https://openalex.org/I150468666"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Lars Ruthotto","raw_affiliation_strings":["Emory Univ., Atlanta, GA (United States)"],"affiliations":[{"raw_affiliation_string":"Emory Univ., Atlanta, GA (United States)","institution_ids":["https://openalex.org/I150468666"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5038235793","display_name":"Jacob B. Schroder","orcid":"https://orcid.org/0000-0002-1076-9206"},"institutions":[{"id":"https://openalex.org/I169521973","display_name":"University of New Mexico","ror":"https://ror.org/05fs6jp91","country_code":"US","type":"education","lineage":["https://openalex.org/I169521973"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Jacob B. Schroder","raw_affiliation_strings":["Univ. of New Mexico, Albuquerque, NM (United States)"],"affiliations":[{"raw_affiliation_string":"Univ. of New Mexico, Albuquerque, NM (United States)","institution_ids":["https://openalex.org/I169521973"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5052883660","display_name":"Eric C. Cyr","orcid":"https://orcid.org/0000-0003-3833-9598"},"institutions":[{"id":"https://openalex.org/I192454743","display_name":"Sandia National Laboratories California","ror":"https://ror.org/058m7ey48","country_code":"US","type":"facility","lineage":["https://openalex.org/I1330989302","https://openalex.org/I1330989302","https://openalex.org/I192454743","https://openalex.org/I198811213","https://openalex.org/I198811213","https://openalex.org/I4210104735"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Eric C. Cyr","raw_affiliation_strings":["Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)"],"affiliations":[{"raw_affiliation_string":"Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)","institution_ids":["https://openalex.org/I192454743"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5023667410","display_name":"Nicolas R. Gauger","orcid":"https://orcid.org/0000-0002-5863-7384"},"institutions":[{"id":"https://openalex.org/I153267046","display_name":"University of Kaiserslautern","ror":"https://ror.org/04zrf7b53","country_code":"DE","type":"education","lineage":["https://openalex.org/I153267046"]}],"countries":["DE"],"is_corresponding":false,"raw_author_name":"Nicolas R. Gauger","raw_affiliation_strings":["Univ. of Kaiserslautern (Germany)"],"affiliations":[{"raw_affiliation_string":"Univ. of Kaiserslautern (Germany)","institution_ids":["https://openalex.org/I153267046"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":5,"corresponding_author_ids":["https://openalex.org/A5044666766"],"corresponding_institution_ids":["https://openalex.org/I153267046"],"apc_list":null,"apc_paid":null,"fwci":6.3788,"has_fulltext":false,"cited_by_count":86,"citation_normalized_percentile":{"value":0.97430442,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":100},"biblio":{"volume":"2","issue":"1","first_page":"1","last_page":"23"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10036","display_name":"Advanced Neural Network Applications","score":0.996399998664856,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10036","display_name":"Advanced Neural Network Applications","score":0.996399998664856,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9926000237464905,"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/T10271","display_name":"Seismic Imaging and Inversion Techniques","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/1908","display_name":"Geophysics"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.8086686134338379},{"id":"https://openalex.org/keywords/speedup","display_name":"Speedup","score":0.7855546474456787},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.709077000617981},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.6060841083526611},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.5786907076835632},{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.545509397983551},{"id":"https://openalex.org/keywords/deep-learning","display_name":"Deep learning","score":0.5188152194023132},{"id":"https://openalex.org/keywords/backpropagation","display_name":"Backpropagation","score":0.4651333689689636},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.453088641166687},{"id":"https://openalex.org/keywords/data-parallelism","display_name":"Data parallelism","score":0.45032185316085815},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.4243309199810028},{"id":"https://openalex.org/keywords/parallelism","display_name":"Parallelism (grammar)","score":0.4177086651325226},{"id":"https://openalex.org/keywords/parallel-computing","display_name":"Parallel computing","score":0.36486971378326416},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.35909324884414673}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.8086686134338379},{"id":"https://openalex.org/C68339613","wikidata":"https://www.wikidata.org/wiki/Q1549489","display_name":"Speedup","level":2,"score":0.7855546474456787},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.709077000617981},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.6060841083526611},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.5786907076835632},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.545509397983551},{"id":"https://openalex.org/C108583219","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep learning","level":2,"score":0.5188152194023132},{"id":"https://openalex.org/C155032097","wikidata":"https://www.wikidata.org/wiki/Q798503","display_name":"Backpropagation","level":3,"score":0.4651333689689636},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.453088641166687},{"id":"https://openalex.org/C61483411","wikidata":"https://www.wikidata.org/wiki/Q3124522","display_name":"Data parallelism","level":3,"score":0.45032185316085815},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4243309199810028},{"id":"https://openalex.org/C2781172179","wikidata":"https://www.wikidata.org/wiki/Q853109","display_name":"Parallelism (grammar)","level":2,"score":0.4177086651325226},{"id":"https://openalex.org/C173608175","wikidata":"https://www.wikidata.org/wiki/Q232661","display_name":"Parallel computing","level":1,"score":0.36486971378326416},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.35909324884414673},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C77088390","wikidata":"https://www.wikidata.org/wiki/Q8513","display_name":"Database","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/19m1247620","is_oa":true,"landing_page_url":"https://doi.org/10.1137/19m1247620","pdf_url":null,"source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematics of Data Science","raw_type":"journal-article"},{"id":"pmh:oai:osti.gov:1618082","is_oa":true,"landing_page_url":"https://www.osti.gov/biblio/1618082","pdf_url":null,"source":{"id":"https://openalex.org/S4306402487","display_name":"OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I139351228","host_organization_name":"Office of Scientific and Technical Information","host_organization_lineage":["https://openalex.org/I139351228"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":null}],"best_oa_location":{"id":"doi:10.1137/19m1247620","is_oa":true,"landing_page_url":"https://doi.org/10.1137/19m1247620","pdf_url":null,"source":{"id":"https://openalex.org/S4210229561","display_name":"SIAM Journal on Mathematics of Data Science","issn_l":"2577-0187","issn":["2577-0187"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Mathematics of Data Science","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1094550926","display_name":null,"funder_award_id":"DMS 1522599, DMS 1751636","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G5211897158","display_name":null,"funder_award_id":"DE-NA0003525","funder_id":"https://openalex.org/F4320332369","funder_display_name":"National Nuclear Security Administration"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320332369","display_name":"National Nuclear Security Administration","ror":"https://ror.org/03sk1we31"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":39,"referenced_works":["https://openalex.org/W1994530392","https://openalex.org/W2006700733","https://openalex.org/W2012630580","https://openalex.org/W2041557582","https://openalex.org/W2059184141","https://openalex.org/W2079522263","https://openalex.org/W2101506489","https://openalex.org/W2105970221","https://openalex.org/W2111870929","https://openalex.org/W2112796928","https://openalex.org/W2126906098","https://openalex.org/W2151317657","https://openalex.org/W2154579312","https://openalex.org/W2158899491","https://openalex.org/W2160815625","https://openalex.org/W2163605009","https://openalex.org/W2168231600","https://openalex.org/W2194775991","https://openalex.org/W2302255633","https://openalex.org/W2524221349","https://openalex.org/W2526749411","https://openalex.org/W2591722035","https://openalex.org/W2612252966","https://openalex.org/W2614984214","https://openalex.org/W2742437467","https://openalex.org/W2756433771","https://openalex.org/W2765729611","https://openalex.org/W2767138068","https://openalex.org/W2919115771","https://openalex.org/W2950344723","https://openalex.org/W2951622387","https://openalex.org/W2953384591","https://openalex.org/W2962911728","https://openalex.org/W2963684275","https://openalex.org/W2964016678","https://openalex.org/W2964075885","https://openalex.org/W3098011980","https://openalex.org/W3105020391","https://openalex.org/W4231109964"],"related_works":["https://openalex.org/W2950520577","https://openalex.org/W1554644772","https://openalex.org/W2003935582","https://openalex.org/W2494130044","https://openalex.org/W3170887803","https://openalex.org/W2963831937","https://openalex.org/W74409296","https://openalex.org/W3209384898","https://openalex.org/W4400951174","https://openalex.org/W1595834484"],"abstract_inverted_index":{"Residual":[0],"neural":[1,10,53],"networks":[2,11],"(ResNets)":[3],"are":[4,172],"a":[5,18,39,57,121,133,162,199],"promising":[6],"class":[7],"of":[8,20,38,51,67,95,103,136,205],"deep":[9,146],"that":[12,140,194,204],"have":[13,84],"shown":[14,85],"excellent":[15],"performance":[16,201],"for":[17,98],"number":[19],"learning":[21],"tasks,":[22],"e.g.,":[23],"image":[24],"classification":[25],"and":[26,100,113,211],"recognition.":[27],"Mathematically,":[28],"ResNet":[29,58],"architectures":[30],"can":[31,59],"be":[32,60],"interpreted":[33],"as":[34,62],"forward":[35,112],"Euler":[36],"discretizations":[37],"nonlinear":[40,123],"initial":[41],"value":[42],"problem":[43,66],"whose":[44],"time-dependent":[45,74],"control":[46,65,76],"variables":[47],"represent":[48],"the":[49,52,68,93,109,117,128,169,183,195],"weights":[50],"network.":[54],"Hence,":[55],"training":[56,102,144,184,200],"cast":[61],"an":[63],"optimal":[64,75],"associated":[69],"dynamical":[70],"system.":[71],"For":[72],"similar":[73,202],"problems":[77],"arising":[78],"in":[79,88,178],"engineering":[80],"applications,":[81],"parallel-in-time":[82],"methods":[83,217],"notable":[86],"improvements":[87],"scalability.":[89],"This":[90,131],"paper":[91],"demonstrates":[92],"use":[94],"those":[96],"techniques":[97],"efficient":[99,159],"effective":[101],"ResNets.":[104],"The":[105,157],"proposed":[106],"algorithms":[107],"replace":[108],"classical":[110],"(sequential)":[111],"backward":[114],"propagation":[115],"through":[116,218],"network":[118,170],"layers":[119,139],"with":[120],"parallel":[122],"multigrid":[124],"iteration":[125],"applied":[126],"to":[127,168,180,203],"layer":[129],"domain.":[130],"adds":[132],"new":[134,196],"dimension":[135],"parallelism":[137],"across":[138],"is":[141],"attractive":[142],"when":[143],"very":[145],"networks.":[147],"From":[148],"this":[149],"basic":[150],"idea,":[151],"we":[152,192],"derive":[153],"multiple":[154],"layer-parallel":[155],"methods.":[156],"most":[158],"version":[160],"employs":[161],"simultaneous":[163],"optimization":[164],"approach":[165,197],"where":[166],"updates":[167],"parameters":[171],"based":[173],"on":[174],"inexact":[175],"gradient":[176],"information":[177],"order":[179],"speed":[181],"up":[182],"process.":[185],"Using":[186],"numerical":[187],"examples":[188],"from":[189],"supervised":[190],"classification,":[191],"demonstrate":[193],"achieves":[198],"traditional":[206],"methods,":[207],"but":[208],"enables":[209],"layer-parallelism":[210],"thus":[212],"provides":[213],"speedup":[214],"over":[215],"layer-serial":[216],"greater":[219],"concurrency.":[220]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":7},{"year":2023,"cited_by_count":13},{"year":2022,"cited_by_count":17},{"year":2021,"cited_by_count":13},{"year":2020,"cited_by_count":22},{"year":2019,"cited_by_count":4}],"updated_date":"2026-03-20T23:20:44.827607","created_date":"2025-10-10T00:00:00"}