{"id":"https://openalex.org/W7214778403","doi":"https://doi.org/10.48550/arxiv.2609.34549","title":"On Parameter Symmetries and Conservation Laws in Gradient Flow","display_name":"On Parameter Symmetries and Conservation Laws in Gradient Flow","publication_year":2026,"publication_date":"2026-09-28","ids":{"openalex":"https://openalex.org/W7214778403","doi":"https://doi.org/10.48550/arxiv.2609.34549"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.34549","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.34549","pdf_url":"https://arxiv.org/pdf/2609.34549","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2609.34549","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5132688201","display_name":"Khang Nguyen","orcid":"https://orcid.org/0000-0003-3471-5533","observed_orcids":["https://orcid.org/0000-0003-3471-5533"]},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Nguyen, Khang","raw_affiliation_strings":["UCLA"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"UCLA","institution_ids":["https://openalex.org/I161318765"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072280192","display_name":"Guido Mont\u00fafar","orcid":"https://orcid.org/0000-0002-0131-2669","observed_orcids":["https://orcid.org/0000-0002-0131-2669"]},"institutions":[{"id":"https://openalex.org/I161318765","display_name":"University of California, Los Angeles","ror":"https://ror.org/046rm7j60","country_code":"US","type":"education","lineage":["https://openalex.org/I161318765"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Mont\u00fafar, Guido","raw_affiliation_strings":["UCLA & MPI MiS"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"UCLA & MPI MiS","institution_ids":["https://openalex.org/I161318765"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I161318765"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"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":0.6942999958992004,"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":0.6942999958992004,"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/T10320","display_name":"Neural Networks and Applications","score":0.26460000872612,"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/T10036","display_name":"Advanced Neural Network Applications","score":0.006200000178068876,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/conservation-laws","display_name":"conservation laws","score":0.996999979019165},{"id":"https://openalex.org/keywords/gradient-flow","display_name":"gradient flow","score":0.984000027179718},{"id":"https://openalex.org/keywords/noether-s-theorem","display_name":"Noether's theorem","score":0.9369999766349792},{"id":"https://openalex.org/keywords/loss-landscape","display_name":"loss landscape","score":0.7329999804496765},{"id":"https://openalex.org/keywords/implicit-bias","display_name":"implicit bias","score":0.8169999718666077},{"id":"https://openalex.org/keywords/neural-network","display_name":"artificial neural networks","score":0.7200000286102295},{"id":"https://openalex.org/keywords/multilayer-networks","display_name":"multilayer networks","score":0.7350000143051147},{"id":"https://openalex.org/keywords/attention-mechanism","display_name":"attention mechanism","score":0.75},{"id":"https://openalex.org/keywords/polynomial-neural-network","display_name":"polynomial neural network","score":0.972000002861023},{"id":"https://openalex.org/keywords/deep-linear-networks","display_name":"deep linear networks","score":0.8379999995231628},{"id":"https://openalex.org/keywords/geometric-framework","display_name":"geometric framework","score":0.5450000166893005}],"concepts":[{"id":"https://openalex.org/C3445786","wikidata":"https://www.wikidata.org/wiki/Q205805","display_name":"Conservation law","level":2,"score":0.8683000206947327},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.7649999856948853},{"id":"https://openalex.org/C122770356","wikidata":"https://www.wikidata.org/wiki/Q1656753","display_name":"Identifiability","level":2,"score":0.623199999332428},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.574400007724762},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.5295000076293945},{"id":"https://openalex.org/C167879884","wikidata":"https://www.wikidata.org/wiki/Q727568","display_name":"Balanced flow","level":2,"score":0.4738999903202057},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.4410000145435333},{"id":"https://openalex.org/C2778012447","wikidata":"https://www.wikidata.org/wiki/Q1034415","display_name":"Scope (computer science)","level":2,"score":0.42730000615119934},{"id":"https://openalex.org/C73586568","wikidata":"https://www.wikidata.org/wiki/Q2600211","display_name":"Parameter space","level":2,"score":0.4251999855041504},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.40059998631477356},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.3822999894618988},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3328999876976013},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3125999867916107},{"id":"https://openalex.org/C33332235","wikidata":"https://www.wikidata.org/wiki/Q18362","display_name":"Theoretical physics","level":1,"score":0.2822999954223633},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.2703999876976013},{"id":"https://openalex.org/C12426560","wikidata":"https://www.wikidata.org/wiki/Q189569","display_name":"Basis (linear algebra)","level":2,"score":0.2703000009059906},{"id":"https://openalex.org/C194583477","wikidata":"https://www.wikidata.org/wiki/Q408891","display_name":"Physical law","level":2,"score":0.2687999904155731},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.25839999318122864},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.2565000057220459},{"id":"https://openalex.org/C77618280","wikidata":"https://www.wikidata.org/wiki/Q1155772","display_name":"Scheme (mathematics)","level":2,"score":0.2529999911785126},{"id":"https://openalex.org/C144237770","wikidata":"https://www.wikidata.org/wiki/Q747534","display_name":"Mathematical economics","level":1,"score":0.2517000138759613}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.34549","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.34549","pdf_url":"https://arxiv.org/pdf/2609.34549","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.34549","doi":"https://doi.org/10.48550/arxiv.2609.34549","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.34549","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.34549","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.34549","pdf_url":"https://arxiv.org/pdf/2609.34549","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"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":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G2023043591","display_name":"Kombinatorische und implizite Methoden f\u00fcr Deep Learning","funder_award_id":"464109215","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G2238847292","display_name":null,"funder_award_id":"2145630","funder_id":"https://openalex.org/F4320337380","funder_display_name":"Division of Mathematical Sciences"},{"id":"https://openalex.org/G442688124","display_name":null,"funder_award_id":"SPP2298","funder_id":"https://openalex.org/F4320320879","funder_display_name":"Deutsche Forschungsgemeinschaft"},{"id":"https://openalex.org/G4712284355","display_name":null,"funder_award_id":"57616814","funder_id":"https://openalex.org/F4320320875","funder_display_name":"Deutscher Akademischer Austauschdienst"},{"id":"https://openalex.org/G5788735763","display_name":"Collaborative Research: RI: Medium: MoDL: Occams Razor in Deep and Physical Learning","funder_award_id":"2212520","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7409962649","display_name":"CAREER: Neural Networks in the Practical Regime","funder_award_id":"2145630","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G8937168971","display_name":"Collaborative Research: AIMing: Synergistic Advancement of AI and Mathematics","funder_award_id":"2522495","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320698","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49"},{"id":"https://openalex.org/F4320320875","display_name":"Deutscher Akademischer Austauschdienst","ror":"https://ror.org/039djdh30"},{"id":"https://openalex.org/F4320320879","display_name":"Deutsche Forschungsgemeinschaft","ror":"https://ror.org/018mejw64"},{"id":"https://openalex.org/F4320332180","display_name":"Defense Advanced Research Projects Agency","ror":"https://ror.org/02caytj08"},{"id":"https://openalex.org/F4320337380","display_name":"Division of Mathematical Sciences","ror":"https://ror.org/051fftw81"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214778403.pdf","grobid_xml":"https://content.openalex.org/works/W7214778403.grobid-xml"},"referenced_works_count":17,"referenced_works":["https://openalex.org/W2064675550","https://openalex.org/W2112796928","https://openalex.org/W2157331557","https://openalex.org/W2298465274","https://openalex.org/W2336006376","https://openalex.org/W2626778328","https://openalex.org/W2950179405","https://openalex.org/W3153281562","https://openalex.org/W4234527189","https://openalex.org/W4292170110","https://openalex.org/W4292872153","https://openalex.org/W4377941395","https://openalex.org/W4415088362","https://openalex.org/W4416435020","https://openalex.org/W4417268274","https://openalex.org/W7161718229","https://openalex.org/W7164152262"],"related_works":["https://openalex.org/W7164152262","https://openalex.org/W3153281562","https://openalex.org/W3111160927","https://openalex.org/W7164998846","https://openalex.org/W7214706051","https://openalex.org/W7161718229","https://openalex.org/W4307652671","https://openalex.org/W4307933733","https://openalex.org/W4416534555","https://openalex.org/W4398230409"],"abstract_inverted_index":{"Parameter":[0],"space":[1],"symmetries":[2,71,96],"and":[3,15,44,83,97,109,115],"conservation":[4,33,74,98],"laws":[5,34,99],"play":[6],"an":[7],"important":[8],"role":[9],"in":[10,25,100],"understanding":[11],"the":[12,42,59,63,67,105],"loss":[13],"landscapes":[14],"implicit":[16],"biases":[17],"of":[18,46,80,95],"neural":[19,113],"networks.":[20,102,119],"Inspired":[21],"by":[22],"Noether's":[23],"theorem":[24],"physics,":[26],"prior":[27],"works":[28],"have":[29],"sought":[30],"to":[31,73,86,107],"derive":[32],"under":[35,69],"gradient":[36],"flow":[37],"from":[38],"parameter":[39],"symmetries,":[40],"but":[41],"scope":[43],"limitations":[45],"this":[47],"connection":[48],"remain":[49],"unclear.":[50],"We":[51,76,103],"develop":[52],"a":[53,78,88],"unified":[54],"geometric":[55],"framework":[56,106],"that":[57],"clarifies":[58],"precise":[60],"relationship":[61],"between":[62],"two":[64],"notions,":[65],"including":[66],"conditions":[68],"which":[70],"correspond":[72],"laws.":[75],"introduce":[77],"notion":[79],"compositional":[81],"identifiability":[82],"use":[84],"it":[85],"establish":[87],"general":[89],"inheritance":[90],"principle":[91],"for":[92],"complete":[93],"characterizations":[94],"multilayer":[101],"apply":[104],"multi-head":[108],"grouped-query":[110],"attention,":[111],"polynomial":[112],"networks,":[114],"square":[116],"deep":[117],"linear":[118]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-30T00:00:00"}
