{"id":"https://openalex.org/W7127441420","doi":"https://doi.org/10.48550/arxiv.2602.01480","title":"Rod Flow: A Continuous-Time Model for Gradient Descent at the Edge of Stability","display_name":"Rod Flow: A Continuous-Time Model for Gradient Descent at the Edge of Stability","publication_year":2026,"publication_date":"2026-02-01","ids":{"openalex":"https://openalex.org/W7127441420","doi":"https://doi.org/10.48550/arxiv.2602.01480"},"language":null,"primary_location":{"id":"pmh:doi:10.48550/arxiv.2602.01480","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","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":null,"any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5026324787","display_name":"Eric Regis","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Regis, Eric","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5052571538","display_name":"Sinho Chewi","orcid":"https://orcid.org/0000-0003-2701-0703"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chewi, Sinho","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5026324787"],"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.7511000037193298,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T11612","display_name":"Stochastic Gradient Optimization Techniques","score":0.7511000037193298,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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":0.11710000038146973,"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/T12611","display_name":"Neural Networks and Reservoir Computing","score":0.01860000006854534,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/ode","display_name":"Ode","score":0.699400007724762},{"id":"https://openalex.org/keywords/gradient-descent","display_name":"Gradient descent","score":0.6740000247955322},{"id":"https://openalex.org/keywords/iterated-function","display_name":"Iterated function","score":0.6696000099182129},{"id":"https://openalex.org/keywords/stability","display_name":"Stability (learning theory)","score":0.6175000071525574},{"id":"https://openalex.org/keywords/simple","display_name":"Simple (philosophy)","score":0.5167999863624573},{"id":"https://openalex.org/keywords/balanced-flow","display_name":"Balanced flow","score":0.48590001463890076},{"id":"https://openalex.org/keywords/enhanced-data-rates-for-gsm-evolution","display_name":"Enhanced Data Rates for GSM Evolution","score":0.4765999913215637},{"id":"https://openalex.org/keywords/flow","display_name":"Flow (mathematics)","score":0.47530001401901245},{"id":"https://openalex.org/keywords/quartic-function","display_name":"Quartic function","score":0.4316999912261963}],"concepts":[{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.699400007724762},{"id":"https://openalex.org/C153258448","wikidata":"https://www.wikidata.org/wiki/Q1199743","display_name":"Gradient descent","level":3,"score":0.6740000247955322},{"id":"https://openalex.org/C140479938","wikidata":"https://www.wikidata.org/wiki/Q5254619","display_name":"Iterated function","level":2,"score":0.6696000099182129},{"id":"https://openalex.org/C112972136","wikidata":"https://www.wikidata.org/wiki/Q7595718","display_name":"Stability (learning theory)","level":2,"score":0.6175000071525574},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5874000191688538},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.5167999863624573},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.48590001463890076},{"id":"https://openalex.org/C162307627","wikidata":"https://www.wikidata.org/wiki/Q204833","display_name":"Enhanced Data Rates for GSM Evolution","level":2,"score":0.4765999913215637},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.47530001401901245},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.4316999912261963},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.3993000090122223},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.39809998869895935},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37279999256134033},{"id":"https://openalex.org/C2776637919","wikidata":"https://www.wikidata.org/wiki/Q624380","display_name":"Descent (aeronautics)","level":2,"score":0.359499990940094},{"id":"https://openalex.org/C145912823","wikidata":"https://www.wikidata.org/wiki/Q113558","display_name":"Dynamics (music)","level":2,"score":0.3587000072002411},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.3450999855995178},{"id":"https://openalex.org/C2781238097","wikidata":"https://www.wikidata.org/wiki/Q175026","display_name":"Object (grammar)","level":2,"score":0.3334999978542328},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.31779998540878296},{"id":"https://openalex.org/C149364088","wikidata":"https://www.wikidata.org/wiki/Q185917","display_name":"Translation (biology)","level":4,"score":0.30889999866485596},{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.3052000105381012},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.30169999599456787},{"id":"https://openalex.org/C176321772","wikidata":"https://www.wikidata.org/wiki/Q1430640","display_name":"Numerical stability","level":3,"score":0.2964000105857849},{"id":"https://openalex.org/C206688291","wikidata":"https://www.wikidata.org/wiki/Q7617819","display_name":"Stochastic gradient descent","level":3,"score":0.2955000102519989},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2888000011444092},{"id":"https://openalex.org/C33962884","wikidata":"https://www.wikidata.org/wiki/Q378637","display_name":"Dynamical system (definition)","level":3,"score":0.2833000123500824},{"id":"https://openalex.org/C79379906","wikidata":"https://www.wikidata.org/wiki/Q3174497","display_name":"Dynamical systems theory","level":2,"score":0.26980000734329224},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.26579999923706055},{"id":"https://openalex.org/C61445026","wikidata":"https://www.wikidata.org/wiki/Q217608","display_name":"Fixed point","level":2,"score":0.25920000672340393}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:doi:10.48550/arxiv.2602.01480","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},{"id":"doi:10.48550/arxiv.2602.01480","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2602.01480","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"pmh:doi:10.48550/arxiv.2602.01480","is_oa":true,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4406922384","display_name":"Open MIND","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"publisher-specific-oa","license_id":"https://openalex.org/licenses/publisher-specific-oa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/11","score":0.6851643323898315,"display_name":"Sustainable cities and communities"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"How":[0],"can":[1],"we":[2,71,138],"understand":[3],"gradient-based":[4],"training":[5],"over":[6,65],"non-convex":[7],"landscapes?":[8],"The":[9],"edge":[10],"of":[11,96,121,162],"stability":[12],"phenomenon,":[13],"introduced":[14],"in":[15,51,152],"Cohen":[16,52],"et":[17,53],"al.":[18,54],"(2021),":[19],"indicates":[20],"that":[21,140],"the":[22,40,46,62,81,119,145],"answer":[23],"is":[24,131],"not":[25],"so":[26],"simple:":[27],"namely,":[28],"gradient":[29,41],"descent":[30],"(GD)":[31],"with":[32,159],"large":[33],"step":[34],"sizes":[35],"often":[36],"diverges":[37],"away":[38],"from":[39,92],"flow.":[42],"In":[43,68],"this":[44,69],"regime,":[45],"\"Central":[47],"Flow\",":[48],"recently":[49],"proposed":[50],"(2025),":[55],"provides":[56],"an":[57,75,100],"accurate":[58],"ODE":[59,77],"approximation":[60],"to":[61,135],"GD":[63,97,111],"dynamics":[64,112],"many":[66],"architectures.":[67],"work,":[70],"propose":[72],"Rod":[73,141],"Flow,":[74],"alternative":[76],"approximation,":[78],"which":[79],"carries":[80],"following":[82],"advantages:":[83],"(1)":[84],"it":[85,108],"rests":[86],"on":[87],"a":[88,93,105,160],"principled":[89],"derivation":[90],"stemming":[91],"physical":[94],"picture":[95],"iterates":[98],"as":[99],"extended":[101],"one-dimensional":[102],"object":[103],"--":[104],"\"rod\";":[106],"(2)":[107],"better":[109],"captures":[110],"for":[113,124],"simple":[114],"toy":[115],"examples":[116],"and":[117,129,133,149],"matches":[118],"accuracy":[120],"Central":[122],"Flow":[123,142],"representative":[125],"neural":[126],"network":[127],"architectures,":[128],"(3)":[130],"explicit":[132],"cheap":[134],"compute.":[136],"Theoretically,":[137],"prove":[139],"correctly":[143],"predicts":[144],"critical":[146],"sharpness":[147],"threshold":[148],"explains":[150],"self-stabilization":[151],"quartic":[153],"potentials.":[154],"We":[155],"validate":[156],"our":[157],"theory":[158],"range":[161],"numerical":[163],"experiments.":[164]},"counts_by_year":[],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2026-02-04T00:00:00"}