{"id":"https://openalex.org/W7219155297","doi":"https://doi.org/10.48550/arxiv.2610.00453","title":"Constructive Neural Network Field Theory: $\u03d5_2^4$ in Finite Volume","display_name":"Constructive Neural Network Field Theory: $\u03d5_2^4$ in Finite Volume","publication_year":2026,"publication_date":"2026-09-30","ids":{"openalex":"https://openalex.org/W7219155297","doi":"https://doi.org/10.48550/arxiv.2610.00453"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.00453","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00453","pdf_url":"https://arxiv.org/pdf/2610.00453","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/2610.00453","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5122977146","display_name":"Samuel Frank","orcid":"https://orcid.org/0000-0003-2597-8155","observed_orcids":["https://orcid.org/0000-0003-2597-8155"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Frank, Samuel","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5122977146"],"corresponding_institution_ids":[],"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/T13234","display_name":"p-Adic Analysis and Mathematical Physics","score":0.5189999938011169,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T13234","display_name":"p-Adic Analysis and Mathematical Physics","score":0.5189999938011169,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.12240000069141388,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10591","display_name":"Theoretical and Computational Physics","score":0.11349999904632568,"subfield":{"id":"https://openalex.org/subfields/3104","display_name":"Condensed Matter Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/constructive-quantum-field-theory","display_name":"constructive quantum field theory","score":0.8579999804496765},{"id":"https://openalex.org/keywords/log-concave-measures","display_name":"log-concave measures","score":0.781000018119812},{"id":"https://openalex.org/keywords/normal-ordering","display_name":"normal ordering","score":0.652999997138977},{"id":"https://openalex.org/keywords/uv-cutoff","display_name":"ultraviolet cutoff","score":0.6639999747276306},{"id":"https://openalex.org/keywords/two-dimensional-field-theory","display_name":"two-dimensional field theory","score":0.5690000057220459},{"id":"https://openalex.org/keywords/three-dimensional-field-theory","display_name":"three-dimensional field theory","score":0.9190000295639038},{"id":"https://openalex.org/keywords/total-variation-distance","display_name":"total variation distance","score":0.5149999856948853}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6086000204086304},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.504800021648407},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4645000100135803},{"id":"https://openalex.org/C9652623","wikidata":"https://www.wikidata.org/wiki/Q190109","display_name":"Field (mathematics)","level":2,"score":0.4311000108718872},{"id":"https://openalex.org/C130432447","wikidata":"https://www.wikidata.org/wiki/Q11420049","display_name":"Quartic function","level":2,"score":0.4187000095844269},{"id":"https://openalex.org/C2781052500","wikidata":"https://www.wikidata.org/wiki/Q2230313","display_name":"Shell (structure)","level":2,"score":0.4108000099658966},{"id":"https://openalex.org/C151201525","wikidata":"https://www.wikidata.org/wiki/Q177239","display_name":"Limit (mathematics)","level":2,"score":0.4106000065803528},{"id":"https://openalex.org/C60718061","wikidata":"https://www.wikidata.org/wiki/Q1414747","display_name":"Momentum (technical analysis)","level":2,"score":0.4007999897003174},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3935000002384186},{"id":"https://openalex.org/C2778217198","wikidata":"https://www.wikidata.org/wiki/Q556977","display_name":"Cutoff","level":2,"score":0.3912999927997589},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.3797999918460846},{"id":"https://openalex.org/C110521144","wikidata":"https://www.wikidata.org/wiki/Q193460","display_name":"Scalar field","level":2,"score":0.36899998784065247},{"id":"https://openalex.org/C2778701210","wikidata":"https://www.wikidata.org/wiki/Q28130034","display_name":"Constructive","level":3,"score":0.3343000113964081},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.32330000400543213},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.3199999928474426},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.31769999861717224},{"id":"https://openalex.org/C131584629","wikidata":"https://www.wikidata.org/wiki/Q4308705","display_name":"Coupling (piping)","level":2,"score":0.31310001015663147},{"id":"https://openalex.org/C197055811","wikidata":"https://www.wikidata.org/wiki/Q207522","display_name":"Probability density function","level":2,"score":0.30959999561309814},{"id":"https://openalex.org/C120174047","wikidata":"https://www.wikidata.org/wiki/Q847073","display_name":"Euclidean distance","level":2,"score":0.30869999527931213},{"id":"https://openalex.org/C21308566","wikidata":"https://www.wikidata.org/wiki/Q7169365","display_name":"Permutation (music)","level":2,"score":0.3086000084877014},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.30300000309944153},{"id":"https://openalex.org/C187399659","wikidata":"https://www.wikidata.org/wiki/Q5500226","display_name":"Free field","level":2,"score":0.2922999858856201},{"id":"https://openalex.org/C7218915","wikidata":"https://www.wikidata.org/wiki/Q1054475","display_name":"Gaussian function","level":3,"score":0.28600001335144043},{"id":"https://openalex.org/C162392398","wikidata":"https://www.wikidata.org/wiki/Q272404","display_name":"Finite set","level":2,"score":0.2856999933719635},{"id":"https://openalex.org/C187268509","wikidata":"https://www.wikidata.org/wiki/Q1402884","display_name":"Field theory (psychology)","level":2,"score":0.2757999897003174},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.27309998869895935},{"id":"https://openalex.org/C2778401447","wikidata":"https://www.wikidata.org/wiki/Q7140637","display_name":"Partition function (quantum field theory)","level":2,"score":0.26899999380111694},{"id":"https://openalex.org/C166785042","wikidata":"https://www.wikidata.org/wiki/Q190391","display_name":"Central limit theorem","level":2,"score":0.2574999928474426},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.2572999894618988}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.00453","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00453","pdf_url":"https://arxiv.org/pdf/2610.00453","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.2610.00453","doi":"https://doi.org/10.48550/arxiv.2610.00453","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.00453","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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.00453","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.00453","pdf_url":"https://arxiv.org/pdf/2610.00453","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":[],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7219155297.pdf","grobid_xml":"https://content.openalex.org/works/W7219155297.grobid-xml"},"referenced_works_count":11,"referenced_works":["https://openalex.org/W4226081363","https://openalex.org/W7123769034","https://openalex.org/W7125385408","https://openalex.org/W7148904960","https://openalex.org/W7159662597","https://openalex.org/W7161033831","https://openalex.org/W7167626497","https://openalex.org/W7203664132","https://openalex.org/W7204165480","https://openalex.org/W7204181962","https://openalex.org/W7214974187"],"related_works":["https://openalex.org/W7204165480","https://openalex.org/W2500814126","https://openalex.org/W7106129993","https://openalex.org/W2159664196","https://openalex.org/W7140874074","https://openalex.org/W4411348016","https://openalex.org/W4394580457","https://openalex.org/W7018111652","https://openalex.org/W6948290878","https://openalex.org/W4392490424"],"abstract_inverted_index":{"Neural":[0],"network":[1,11,32,47,245],"field":[2,7,33,59,242],"theory":[3,8,34],"specifies":[4],"a":[5,10,14,44,62,93,97,135,241,244,256],"Euclidean":[6],"by":[9,101],"architecture":[12],"and":[13,35,72,83,116,123,142,145,176,212,216,250],"density":[15,99,127],"on":[16],"its":[17],"parameters,":[18],"so":[19,263],"that":[20,108,166],"correlation":[21],"functions":[22],"are":[23,85,152,193],"integrals":[24],"over":[25,163],"those":[26,149,230],"parameters.":[27],"We":[28],"introduce":[29],"constructive":[30],"neural":[31],"use":[36],"it":[37,266],"to":[38,154,171,229,261,267,269],"realize":[39],"finite-volume":[40,234],"scalar":[41],"$\u03d5^4_2$":[42,235],"as":[43],"limit":[45],"of":[46,54,65,120,148,186,206,223,231,239],"measures.":[48],"For":[49,106],"each":[50],"dyadic":[51],"momentum":[52,248],"shell":[53,114,133,150,167,254],"the":[55,58,76,81,102,113,121,125,146,155,172,177,187,195,210,220,232],"free":[56],"propagator,":[57],"is":[60,92,128,185,258],"given":[61],"separate":[63],"set":[64],"neurons":[66],"with":[67,112],"Gaussian":[68,98],"coefficients,":[69],"uniform":[70],"phases,":[71,124],"frequencies":[73,82,122],"drawn":[74],"from":[75],"shell's":[77,90],"spectral":[78],"density.":[79],"Once":[80],"phases":[84],"fixed,":[86],"integrating":[87,251],"out":[88,252],"one":[89,253],"coefficients":[91],"finite-dimensional":[94],"integral":[95],"against":[96],"reweighted":[100,126],"Wick-ordered":[103],"quartic":[104],"interaction.":[105],"widths":[107],"grow":[109],"fast":[110],"enough":[111],"index":[115],"almost":[117],"every":[118,132],"draw":[119],"uniformly":[129],"log-concave":[130],"for":[131,201],"above":[134],"threshold,":[136],"at":[137,182,255],"any":[138,202],"coupling":[139],"in":[140,217],"two":[141,159],"three":[143],"dimensions,":[144,160],"fluctuations":[147],"fields":[151,227],"held":[153],"free-field":[156],"scale.":[157],"In":[158],"an":[161],"induction":[162],"scales":[164],"shows":[165],"$j$":[168],"contributes":[169],"$O(2^{-2j}j^3)$":[170],"log":[173],"partition":[174],"function,":[175],"total-variation":[178],"distance":[179],"between":[180],"marginals":[181],"successive":[183],"cutoffs":[184],"same":[188],"order.":[189],"Because":[190],"these":[191],"contributions":[192],"summable,":[194],"ultraviolet":[196,211],"cutoff":[197],"can":[198],"be":[199],"removed":[200],"suitable":[203],"fixed":[204],"sequence":[205],"widths.":[207],"Almost":[208],"surely,":[209],"infinite-width":[213],"limits":[214],"commute,":[215],"either":[218],"order":[219],"joint":[221],"laws":[222],"finitely":[224],"many":[225],"smeared":[226],"converge":[228],"standard":[233],"measure.":[236],"The":[237],"method":[238],"parametrizing":[240],"via":[243],"organized":[246],"into":[247],"shells":[249],"time":[257],"not":[259],"specific":[260],"$\u03d5^4_2$,":[262],"we":[264],"expect":[265],"extend":[268],"other":[270],"theories.":[271]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-10-04T00:00:00"}