{"id":"https://openalex.org/W7167046962","doi":"https://doi.org/10.48550/arxiv.2607.00676","title":"Penalty-Free Natural Deep Ritz Method Based on de Rham Complex for High-Dimensional Dirichlet Boundary Value Problems","display_name":"Penalty-Free Natural Deep Ritz Method Based on de Rham Complex for High-Dimensional Dirichlet Boundary Value Problems","publication_year":2026,"publication_date":"2026-07-01","ids":{"openalex":"https://openalex.org/W7167046962","doi":"https://doi.org/10.48550/arxiv.2607.00676"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2607.00676","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.00676","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,"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"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2607.00676","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101589948","display_name":"Jiarong Chen","orcid":"https://orcid.org/0000-0001-7920-4249"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Chen, Jiarong","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5139938782","display_name":"Xia Ji","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ji, Xia","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5139941617","display_name":"Haijun Yu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Yu, Haijun","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5139864429","display_name":"Shuo Zhang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Zhang, Shuo","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":[],"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":0.7386000156402588,"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":0.7386000156402588,"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/T11804","display_name":"Quantum many-body systems","score":0.04340000078082085,"subfield":{"id":"https://openalex.org/subfields/3107","display_name":"Atomic and Molecular Physics, and Optics"},"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.0215000007301569,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.617900013923645},{"id":"https://openalex.org/keywords/antisymmetric-relation","display_name":"Antisymmetric relation","score":0.5753999948501587},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.5016000270843506},{"id":"https://openalex.org/keywords/dirichlet-distribution","display_name":"Dirichlet distribution","score":0.48660001158714294},{"id":"https://openalex.org/keywords/penalty-method","display_name":"Penalty method","score":0.44589999318122864},{"id":"https://openalex.org/keywords/dirichlet-boundary-condition","display_name":"Dirichlet boundary condition","score":0.40369999408721924},{"id":"https://openalex.org/keywords/scalar","display_name":"Scalar (mathematics)","score":0.3903000056743622}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6297000050544739},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.617900013923645},{"id":"https://openalex.org/C152401794","wikidata":"https://www.wikidata.org/wiki/Q583760","display_name":"Antisymmetric relation","level":2,"score":0.5753999948501587},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5016000270843506},{"id":"https://openalex.org/C169214877","wikidata":"https://www.wikidata.org/wiki/Q981016","display_name":"Dirichlet distribution","level":3,"score":0.48660001158714294},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.475600004196167},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.47429999709129333},{"id":"https://openalex.org/C6180225","wikidata":"https://www.wikidata.org/wiki/Q3411771","display_name":"Penalty method","level":2,"score":0.44589999318122864},{"id":"https://openalex.org/C110167270","wikidata":"https://www.wikidata.org/wiki/Q1193699","display_name":"Dirichlet boundary condition","level":3,"score":0.40369999408721924},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.3903000056743622},{"id":"https://openalex.org/C13626590","wikidata":"https://www.wikidata.org/wiki/Q318754","display_name":"Summation by parts","level":2,"score":0.3517000079154968},{"id":"https://openalex.org/C2779560616","wikidata":"https://www.wikidata.org/wiki/Q1192869","display_name":"Dirichlet problem","level":3,"score":0.3237000107765198},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.3199000060558319},{"id":"https://openalex.org/C21575042","wikidata":"https://www.wikidata.org/wiki/Q905132","display_name":"Ritz method","level":3,"score":0.31540000438690186},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.3077000081539154},{"id":"https://openalex.org/C31836371","wikidata":"https://www.wikidata.org/wiki/Q1856609","display_name":"Scalar potential","level":2,"score":0.30149999260902405},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3003999888896942},{"id":"https://openalex.org/C103302479","wikidata":"https://www.wikidata.org/wiki/Q1227685","display_name":"Dirichlet conditions","level":5,"score":0.25040000677108765}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2607.00676","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.00676","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,"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":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2607.00676","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2607.00676","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,"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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Deep":[0,33],"neural":[1],"networks":[2],"show":[3,155],"great":[4],"promise":[5],"for":[6,51,108,185],"high-dimensional":[7],"PDEs,":[8],"yet":[9],"enforcing":[10],"essential":[11],"boundary":[12,66,110,196],"conditions":[13],"remains":[14],"challenging,":[15],"especially":[16],"as":[17],"penalty":[18,111,160,187],"parameters":[19],"require":[20],"problem-specific":[21],"retuning":[22],"with":[23,101],"increasing":[24],"dimensionality.":[25],"In":[26],"this":[27],"work,":[28],"we":[29],"extend":[30],"the":[31,59,106,143,165,199],"Natural":[32],"Ritz":[34],"Method":[35],"(NatDRM)":[36],"[H.":[37],"Yu":[38],"and":[39,63,80,127,136,171,189,195],"S.":[40],"Zhang,":[41],"J.":[42],"Comput.":[43],"Phys.,":[44],"537":[45],"(2025)]":[46],"to":[47,123,133,153],"a":[48,109,128],"unified":[49],"framework":[50],"all":[52],"dimensions":[53],"$d":[54,86],"\\geq":[55,87],"2$":[56],"based":[57],"on":[58,71,149],"de":[60],"Rham":[61],"complex":[62],"its":[64],"penalty-free":[65],"decomposition:":[67],"curl-type":[68],"operators":[69],"act":[70],"scalar":[72],"potentials":[73,77,84],"in":[74,78,85,173,179],"2D,":[75],"vector":[76],"3D,":[79],"antisymmetric":[81],"second-order":[82],"tensor":[83],"4$,":[88],"respectively.":[89],"This":[90],"method":[91],"converts":[92],"Dirichlet":[93],"constraints":[94],"into":[95],"three":[96],"coupled":[97],"natural":[98],"(Neumann-type)":[99],"subproblems":[100],"corresponding":[102],"Ritz-type":[103],"losses,":[104,118],"eliminating":[105],"need":[107],"parameter":[112],"$\u03b2$.":[113],"We":[114],"derive":[115],"dimension-unified":[116],"discrete":[117],"lightweight":[119],"boundary-based":[120],"gauge-fixing":[121],"regularizations":[122],"resolve":[124],"curl-kernel":[125],"non-uniqueness,":[126],"joint":[129],"training":[130],"procedure;":[131],"extensions":[132],"variable-coefficient":[134],"elliptic":[135],"semilinear":[137],"Poisson":[138],"problems":[139],"are":[140],"formulated":[141],"at":[142],"first":[144],"subproblem":[145],"level.":[146],"Numerical":[147],"experiments":[148],"smooth":[150],"benchmarks":[151],"up":[152],"6D":[154,180],"that":[156],"NatDRM,":[157],"without":[158],"any":[159],"tuning,":[161],"matches":[162],"or":[163],"exceeds":[164],"accuracy":[166],"of":[167,193,202],"optimally":[168],"tuned":[169],"DRM":[170,183],"PINN":[172],"most":[174,186],"cases.":[175],"It":[176],"converges":[177],"stably":[178],"where":[181],"penalized":[182],"fails":[184],"values,":[188],"exhibits":[190],"synchronous":[191],"decay":[192],"interior":[194],"errors,":[197],"resolving":[198],"inherent":[200],"imbalance":[201],"penalty-based":[203],"methods.":[204]},"counts_by_year":[],"updated_date":"2026-07-28T07:46:37.118299","created_date":"2026-07-03T00:00:00"}
