{"id":"https://openalex.org/W4391243951","doi":"https://doi.org/10.1007/s11424-024-3294-x","title":"NSNO: Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium","display_name":"NSNO: Neumann Series Neural Operator for Solving Helmholtz Equations in Inhomogeneous Medium","publication_year":2024,"publication_date":"2024-01-25","ids":{"openalex":"https://openalex.org/W4391243951","doi":"https://doi.org/10.1007/s11424-024-3294-x"},"language":"en","primary_location":{"id":"doi:10.1007/s11424-024-3294-x","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s11424-024-3294-x","pdf_url":null,"source":{"id":"https://openalex.org/S45848134","display_name":"Journal of Systems Science and Complexity","issn_l":"1009-6124","issn":["1009-6124","1559-7067"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Systems Science and Complexity","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5059930078","display_name":"Fukai Chen","orcid":null},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Fukai Chen","raw_affiliation_strings":["Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100655683","display_name":"Ziyang Liu","orcid":"https://orcid.org/0000-0001-6828-7587"},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ziyang Liu","raw_affiliation_strings":["Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5013455593","display_name":"Guochang Lin","orcid":null},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Guochang Lin","raw_affiliation_strings":["Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5101962646","display_name":"Junqing Chen","orcid":"https://orcid.org/0000-0002-8262-1526"},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Junqing Chen","raw_affiliation_strings":["Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5077724126","display_name":"Zuoqiang Shi","orcid":"https://orcid.org/0000-0002-9122-0302"},"institutions":[{"id":"https://openalex.org/I4403928416","display_name":"Beijing Institute of Mathematical Sciences and Applications","ror":"https://ror.org/05t6hvr95","country_code":null,"type":"education","lineage":["https://openalex.org/I4403928416","https://openalex.org/I99065089"]},{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Zuoqiang Shi","raw_affiliation_strings":["Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China","Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Yanqi Lake Beijing Institute of Mathematical Sciences and Applications, Beijing, 101408, China","institution_ids":["https://openalex.org/I4403928416"]},{"raw_affiliation_string":"Yau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China","institution_ids":["https://openalex.org/I99065089"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5077724126"],"corresponding_institution_ids":["https://openalex.org/I4403928416","https://openalex.org/I99065089"],"apc_list":{"value":2890,"currency":"USD","value_usd":2890},"apc_paid":null,"fwci":1.6802,"has_fulltext":false,"cited_by_count":5,"citation_normalized_percentile":{"value":0.78839704,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"37","issue":"2","first_page":"413","last_page":"440"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10271","display_name":"Seismic Imaging and Inversion Techniques","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/1908","display_name":"Geophysics"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10271","display_name":"Seismic Imaging and Inversion Techniques","score":0.9966999888420105,"subfield":{"id":"https://openalex.org/subfields/1908","display_name":"Geophysics"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10688","display_name":"Image and Signal Denoising Methods","score":0.9965999722480774,"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"}},{"id":"https://openalex.org/T11757","display_name":"Seismic Waves and Analysis","score":0.9943000078201294,"subfield":{"id":"https://openalex.org/subfields/1908","display_name":"Geophysics"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/neumann-series","display_name":"Neumann series","score":0.796819806098938},{"id":"https://openalex.org/keywords/complex-system","display_name":"Complex system","score":0.7844022512435913},{"id":"https://openalex.org/keywords/helmholtz-free-energy","display_name":"Helmholtz free energy","score":0.6542514562606812},{"id":"https://openalex.org/keywords/operator","display_name":"Operator (biology)","score":0.6305769681930542},{"id":"https://openalex.org/keywords/series","display_name":"Series (stratigraphy)","score":0.6239017248153687},{"id":"https://openalex.org/keywords/von-neumanns-theorem","display_name":"Von Neumann's theorem","score":0.5599058270454407},{"id":"https://openalex.org/keywords/helmholtz-equation","display_name":"Helmholtz equation","score":0.5379299521446228},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.4655820429325104},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.44484713673591614},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4334242343902588},{"id":"https://openalex.org/keywords/von-neumann-architecture","display_name":"Von Neumann architecture","score":0.4236529469490051},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.41940316557884216},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.309525728225708},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.26216626167297363},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.18438255786895752},{"id":"https://openalex.org/keywords/multiplication-operator","display_name":"Multiplication operator","score":0.16324108839035034},{"id":"https://openalex.org/keywords/hilbert-space","display_name":"Hilbert space","score":0.14243444800376892},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.13871774077415466},{"id":"https://openalex.org/keywords/geology","display_name":"Geology","score":0.09713998436927795},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.08680564165115356}],"concepts":[{"id":"https://openalex.org/C137343772","wikidata":"https://www.wikidata.org/wiki/Q584910","display_name":"Neumann series","level":2,"score":0.796819806098938},{"id":"https://openalex.org/C47822265","wikidata":"https://www.wikidata.org/wiki/Q854457","display_name":"Complex system","level":2,"score":0.7844022512435913},{"id":"https://openalex.org/C27592594","wikidata":"https://www.wikidata.org/wiki/Q865821","display_name":"Helmholtz free energy","level":2,"score":0.6542514562606812},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.6305769681930542},{"id":"https://openalex.org/C143724316","wikidata":"https://www.wikidata.org/wiki/Q312468","display_name":"Series (stratigraphy)","level":2,"score":0.6239017248153687},{"id":"https://openalex.org/C21261745","wikidata":"https://www.wikidata.org/wiki/Q7941491","display_name":"Von Neumann's theorem","level":4,"score":0.5599058270454407},{"id":"https://openalex.org/C18591234","wikidata":"https://www.wikidata.org/wiki/Q860615","display_name":"Helmholtz equation","level":3,"score":0.5379299521446228},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.4655820429325104},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.44484713673591614},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4334242343902588},{"id":"https://openalex.org/C80469333","wikidata":"https://www.wikidata.org/wiki/Q189088","display_name":"Von Neumann architecture","level":2,"score":0.4236529469490051},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.41940316557884216},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.309525728225708},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.26216626167297363},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.18438255786895752},{"id":"https://openalex.org/C82916341","wikidata":"https://www.wikidata.org/wiki/Q6934990","display_name":"Multiplication operator","level":3,"score":0.16324108839035034},{"id":"https://openalex.org/C62799726","wikidata":"https://www.wikidata.org/wiki/Q190056","display_name":"Hilbert space","level":2,"score":0.14243444800376892},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.13871774077415466},{"id":"https://openalex.org/C127313418","wikidata":"https://www.wikidata.org/wiki/Q1069","display_name":"Geology","level":0,"score":0.09713998436927795},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.08680564165115356},{"id":"https://openalex.org/C86339819","wikidata":"https://www.wikidata.org/wiki/Q407384","display_name":"Transcription factor","level":3,"score":0.0},{"id":"https://openalex.org/C185592680","wikidata":"https://www.wikidata.org/wiki/Q2329","display_name":"Chemistry","level":0,"score":0.0},{"id":"https://openalex.org/C158448853","wikidata":"https://www.wikidata.org/wiki/Q425218","display_name":"Repressor","level":4,"score":0.0},{"id":"https://openalex.org/C104317684","wikidata":"https://www.wikidata.org/wiki/Q7187","display_name":"Gene","level":2,"score":0.0},{"id":"https://openalex.org/C55493867","wikidata":"https://www.wikidata.org/wiki/Q7094","display_name":"Biochemistry","level":1,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0},{"id":"https://openalex.org/C151730666","wikidata":"https://www.wikidata.org/wiki/Q7205","display_name":"Paleontology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1007/s11424-024-3294-x","is_oa":false,"landing_page_url":"https://doi.org/10.1007/s11424-024-3294-x","pdf_url":null,"source":{"id":"https://openalex.org/S45848134","display_name":"Journal of Systems Science and Complexity","issn_l":"1009-6124","issn":["1009-6124","1559-7067"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319900","host_organization_name":"Springer Science+Business Media","host_organization_lineage":["https://openalex.org/P4310319900","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Springer Science+Business Media","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Journal of Systems Science and Complexity","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/9","display_name":"Industry, innovation and infrastructure","score":0.4399999976158142}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W1523113578","https://openalex.org/W1559172566","https://openalex.org/W1901129140","https://openalex.org/W2001019613","https://openalex.org/W2007339694","https://openalex.org/W2030729182","https://openalex.org/W2033106066","https://openalex.org/W2051434435","https://openalex.org/W2068928603","https://openalex.org/W2069143585","https://openalex.org/W2070299075","https://openalex.org/W2080212747","https://openalex.org/W2125916088","https://openalex.org/W2137983211","https://openalex.org/W2140153041","https://openalex.org/W2884001105","https://openalex.org/W2884436604","https://openalex.org/W2894120608","https://openalex.org/W2899283552","https://openalex.org/W2964227007","https://openalex.org/W2979786244","https://openalex.org/W3014382468","https://openalex.org/W3097938572","https://openalex.org/W3108410618","https://openalex.org/W3196492281","https://openalex.org/W3201666041","https://openalex.org/W3202040256","https://openalex.org/W4214759498","https://openalex.org/W4221019179","https://openalex.org/W4235898503","https://openalex.org/W4320519852","https://openalex.org/W6600005967","https://openalex.org/W6602670149"],"related_works":["https://openalex.org/W4365420316","https://openalex.org/W3193896743","https://openalex.org/W2493263495","https://openalex.org/W1569709387","https://openalex.org/W1587032931","https://openalex.org/W2079504277","https://openalex.org/W2962864206","https://openalex.org/W1596866380","https://openalex.org/W2963309482","https://openalex.org/W2156278396"],"abstract_inverted_index":null,"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":2},{"year":2024,"cited_by_count":1}],"updated_date":"2026-07-23T08:03:31.855105","created_date":"2025-10-10T00:00:00"}