{"id":"https://openalex.org/W7214588971","doi":"https://doi.org/10.48550/arxiv.2609.30332","title":"Physics-Informed Neural Networks for Static Black-Hole Exterior Metrics: Charge and Cosmological-Constant Sweeps","display_name":"Physics-Informed Neural Networks for Static Black-Hole Exterior Metrics: Charge and Cosmological-Constant Sweeps","publication_year":2026,"publication_date":"2026-09-24","ids":{"openalex":"https://openalex.org/W7214588971","doi":"https://doi.org/10.48550/arxiv.2609.30332"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.30332","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30332","pdf_url":"https://arxiv.org/pdf/2609.30332","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/2609.30332","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5153199511","display_name":"Huan Jin","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jin, Huan","raw_affiliation_strings":["School of Information Engineering , Jiangxi Institute of Technology , Nanchang , Jiangxi 330098 , PR China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Information Engineering , Jiangxi Institute of Technology , Nanchang , Jiangxi 330098 , PR China","institution_ids":[]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5144168475","display_name":"Fei Wu","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Wu, Fei","raw_affiliation_strings":["School of Information Engineering , Jiangxi Institute of Technology , Nanchang , Jiangxi 330098 , PR China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Information Engineering , Jiangxi Institute of Technology , Nanchang , Jiangxi 330098 , PR China","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5153292805","display_name":"Fei Xue","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xue, Fei","raw_affiliation_strings":["School of Information Engineering , Jiangxi Institute of Technology , Nanchang , Jiangxi 330098 , PR China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Information Engineering , Jiangxi Institute of Technology , Nanchang , Jiangxi 330098 , PR China","institution_ids":[]}]}],"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":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/T11206","display_name":"Model Reduction and Neural Networks","score":0.5536999702453613,"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":0.5536999702453613,"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/T10025","display_name":"Black Holes and Theoretical Physics","score":0.1737000048160553,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy 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/T10463","display_name":"Pulsars and Gravitational Waves Research","score":0.06369999796152115,"subfield":{"id":"https://openalex.org/subfields/3103","display_name":"Astronomy and Astrophysics"},"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/physics-informed-neural-networks","display_name":"physics-informed neural networks","score":0.9879999756813049},{"id":"https://openalex.org/keywords/static-spherically-symmetric-spacetimes","display_name":"static spherically symmetric spacetimes","score":0.8809999823570251},{"id":"https://openalex.org/keywords/einstein-field-equations","display_name":"Einstein field equations","score":0.8799999952316284},{"id":"https://openalex.org/keywords/charged-black-holes","display_name":"charged black holes","score":0.8270000219345093},{"id":"https://openalex.org/keywords/cosmological-constant","display_name":"cosmological constant","score":0.968999981880188},{"id":"https://openalex.org/keywords/asymptotic-boundary-conditions","display_name":"asymptotic boundary conditions","score":0.6320000290870667},{"id":"https://openalex.org/keywords/residual-loss","display_name":"residual loss","score":0.35899999737739563}],"concepts":[{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.6347000002861023},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.5688999891281128},{"id":"https://openalex.org/C97256817","wikidata":"https://www.wikidata.org/wiki/Q1462316","display_name":"Spurious relationship","level":2,"score":0.5246000289916992},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5116000175476074},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.49129998683929443},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.45660001039505005},{"id":"https://openalex.org/C41608201","wikidata":"https://www.wikidata.org/wiki/Q980509","display_name":"Embedding","level":2,"score":0.44780001044273376},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.43560001254081726},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.4198000133037567},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.4099000096321106},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.38760000467300415},{"id":"https://openalex.org/C176217482","wikidata":"https://www.wikidata.org/wiki/Q860554","display_name":"Metric (unit)","level":2,"score":0.3714999854564667},{"id":"https://openalex.org/C34862557","wikidata":"https://www.wikidata.org/wiki/Q178985","display_name":"Ode","level":2,"score":0.3684999942779541},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.358599990606308},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.31709998846054077},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.31459999084472656},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3052999973297119},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2989000082015991},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.29269999265670776},{"id":"https://openalex.org/C178635117","wikidata":"https://www.wikidata.org/wiki/Q747499","display_name":"RADIUS","level":2,"score":0.27869999408721924},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2782000005245209},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.2606000006198883},{"id":"https://openalex.org/C186633575","wikidata":"https://www.wikidata.org/wiki/Q845060","display_name":"Maxima and minima","level":2,"score":0.25619998574256897},{"id":"https://openalex.org/C18762648","wikidata":"https://www.wikidata.org/wiki/Q42213","display_name":"Work (physics)","level":2,"score":0.2524999976158142}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.30332","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30332","pdf_url":"https://arxiv.org/pdf/2609.30332","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.2609.30332","doi":"https://doi.org/10.48550/arxiv.2609.30332","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.30332","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:2609.30332","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.30332","pdf_url":"https://arxiv.org/pdf/2609.30332","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":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7214588971.pdf","grobid_xml":"https://content.openalex.org/works/W7214588971.grobid-xml"},"referenced_works_count":9,"referenced_works":["https://openalex.org/W2767526854","https://openalex.org/W2998763638","https://openalex.org/W3118861991","https://openalex.org/W3123377259","https://openalex.org/W3194436063","https://openalex.org/W4225717750","https://openalex.org/W4287671439","https://openalex.org/W4386794612","https://openalex.org/W4391629689"],"related_works":["https://openalex.org/W7202212463","https://openalex.org/W7220674555","https://openalex.org/W4415145775","https://openalex.org/W4404211397","https://openalex.org/W7167687047","https://openalex.org/W4390601173","https://openalex.org/W7223148958","https://openalex.org/W3193146130","https://openalex.org/W7164163069","https://openalex.org/W4313304141"],"abstract_inverted_index":{"We":[0],"apply":[1],"physics-informed":[2],"neural":[3],"networks":[4],"(PINNs)":[5],"to":[6,105],"recover":[7],"the":[8,39,61,65,74,90,108,115,128,139,165,169,196],"time-time":[9],"component":[10],"of":[11,70,114,183,195],"static,":[12],"spherically":[13],"symmetric":[14],"black-hole":[15],"exterior":[16,41],"metrics":[17],"from":[18],"a":[19,43,84],"reduced":[20],"ordinary":[21],"differential":[22],"equation":[23],"(ODE).":[24],"Inspired":[25],"by":[26,51,135],"recent":[27],"work":[28],"on":[29],"solving":[30],"Einstein":[31],"field":[32,201],"equations":[33],"with":[34,79,177],"deep":[35],"learning~\\cite{Li2023},":[36],"we":[37,82,144],"encode":[38],"vacuum/charged":[40],"through":[42],"residual":[44],"loss":[45],"and":[46,111,126,151,186],"an":[47],"asymptotic":[48],"boundary":[49],"constraint---not":[50],"embedding":[52],"analytic":[53,166],"metric":[54],"terms":[55],"such":[56],"as":[57],"$2M/r$":[58],"directly":[59],"into":[60,77],"network":[62,88],"output.":[63],"Unlike":[64],"distributed":[66],"PINN":[67],"(DPINN)":[68],"strategy":[69],"Ref.~\\cite{Li2023},":[71],"which":[72],"partitions":[73],"radial":[75],"domain":[76,136],"subdomains":[78],"separate":[80],"networks,":[81],"employ":[83],"\\emph{unified}":[85],"fully":[86],"connected":[87],"over":[89],"entire":[91],"interval":[92],"$[r_{\\min},r_{\\max}]$,":[93],"avoiding":[94],"spurious":[95],"jumps":[96],"at":[97,119,142,149,154],"subdomain":[98],"interfaces.":[99],"Moreover,":[100],"whereas":[101],"Ref.~\\cite{Li2023}":[102],"restricts":[103],"training":[104],"$r\\in(10,300)$---well":[106],"outside":[107],"steep":[109],"$1/r$":[110],"$1/r^2$":[112],"curvature":[113],"inner":[116],"exterior---we":[117],"begin":[118],"$r_{\\min}=10^{-2}M$,":[120],"spanning":[121],"three":[122,174],"decades":[123],"in":[124],"radius":[125],"covering":[127],"strongly":[129],"varying":[130],"region":[131],"that":[132],"DPINNs":[133],"sidestep":[134],"truncation.":[137],"Holding":[138],"mass":[140],"fixed":[141,178],"$M=1$,":[143],"sweep":[145,152],"electric":[146],"charge":[147],"$Q\\in\\{0,0.5,1.0,1.1\\}$":[148],"$\u039b=0$":[150],"$\u039b\\in\\{0,0.1,-0.1\\}$":[153],"$Q=0.5$.":[155],"All":[156],"configurations":[157],"achieve":[158],"relative":[159,181],"$\\mathcal{L}_2$":[160],"errors":[161,182],"below":[162],"$6\\%$":[163],"against":[164],"reference.":[167],"For":[168],"representative":[170],"case":[171],"$Q=0.5$,":[172],"$\u039b=0.1$,":[173],"independent":[175],"trainings":[176],"seeds":[179],"yield":[180],"$3.89\\%$,":[184],"$1.08\\%$,":[185],"$2.76\\%$":[187],"(mean":[188],"$2.58\\%$,":[189],"standard":[190],"deviation":[191],"$1.41\\%$),":[192],"demonstrating":[193],"robustness":[194],"mesh-free":[197],"approach":[198],"without":[199],"labeled":[200],"data.":[202]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-29T00:00:00"}