{"id":"https://openalex.org/W7154107343","doi":"https://doi.org/10.3390/math14081275","title":"Fourier Neural Operator for Turbine Wake Flow Prediction with Out-of-Distribution Generalization","display_name":"Fourier Neural Operator for Turbine Wake Flow Prediction with Out-of-Distribution Generalization","publication_year":2026,"publication_date":"2026-04-11","ids":{"openalex":"https://openalex.org/W7154107343","doi":"https://doi.org/10.3390/math14081275"},"language":"en","primary_location":{"id":"doi:10.3390/math14081275","doi":"https://doi.org/10.3390/math14081275","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14081275","pdf_url":"https://www.mdpi.com/2227-7390/14/8/1275/pdf?version=1775917125","source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/2227-7390/14/8/1275/pdf?version=1775917125","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5068659166","display_name":"Shan Ai","orcid":null,"observed_orcids":[]},"institutions":[{"id":"https://openalex.org/I157773358","display_name":"Sun Yat-sen University","ror":"https://ror.org/0064kty71","country_code":"CN","type":"education","lineage":["https://openalex.org/I157773358"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Shan Ai","raw_affiliation_strings":["School of Marine Engineering and Technology, Sun Yat-sen University, Guangzhou 510275, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Marine Engineering and Technology, Sun Yat-sen University, Guangzhou 510275, China","institution_ids":["https://openalex.org/I157773358"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5102081338","display_name":"Chao Hu","orcid":"https://orcid.org/0009-0006-0507-7391","observed_orcids":["https://orcid.org/0009-0006-0507-7391"]},"institutions":[{"id":"https://openalex.org/I157773358","display_name":"Sun Yat-sen University","ror":"https://ror.org/0064kty71","country_code":"CN","type":"education","lineage":["https://openalex.org/I157773358"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Chao Hu","raw_affiliation_strings":["School of Marine Engineering and Technology, Sun Yat-sen University, Guangzhou 510275, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Marine Engineering and Technology, Sun Yat-sen University, Guangzhou 510275, China","institution_ids":["https://openalex.org/I157773358"]}]},{"author_position":"last","author":{"id":null,"display_name":"Yong Ma","orcid":"https://orcid.org/0000-0001-6663-3394","observed_orcids":["https://orcid.org/0000-0001-6663-3394"]},"institutions":[{"id":"https://openalex.org/I157773358","display_name":"Sun Yat-sen University","ror":"https://ror.org/0064kty71","country_code":"CN","type":"education","lineage":["https://openalex.org/I157773358"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Yong Ma","raw_affiliation_strings":["School of Marine Engineering and Technology, Sun Yat-sen University, Guangzhou 510275, China"],"raw_orcid":"https://orcid.org/0000-0001-6663-3394","affiliations":[{"raw_affiliation_string":"School of Marine Engineering and Technology, Sun Yat-sen University, Guangzhou 510275, China","institution_ids":["https://openalex.org/I157773358"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I157773358"],"apc_list":{"value":2600,"currency":"CHF","value_usd":3137},"apc_paid":{"value":2600,"currency":"CHF","value_usd":3137},"fwci":3.3465,"has_fulltext":true,"cited_by_count":1,"citation_normalized_percentile":{"value":0.91171047,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":93,"max":96},"biblio":{"volume":"14","issue":"8","first_page":"1275","last_page":"1275"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9916999936103821,"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":0.9916999936103821,"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/T10680","display_name":"Wind Energy Research and Development","score":0.005900000222027302,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10944","display_name":"Turbomachinery Performance and Optimization","score":0.00039999998989515007,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"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/fourier-neural-operator","display_name":"Fourier neural operator","score":0.9679999947547913},{"id":"https://openalex.org/keywords/out-of-distribution-generalization","display_name":"out-of-distribution generalization","score":0.9509999752044678},{"id":"https://openalex.org/keywords/tip-speed-ratio","display_name":"tip speed ratio","score":0.8479999899864197},{"id":"https://openalex.org/keywords/uncertainty-quantification","display_name":"uncertainty quantification","score":0.902999997138977},{"id":"https://openalex.org/keywords/monte-carlo-dropout","display_name":"Monte Carlo dropout","score":0.8529999852180481},{"id":"https://openalex.org/keywords/lipschitz-regularization","display_name":"Lipschitz regularization","score":0.9020000100135803},{"id":"https://openalex.org/keywords/tidal-stream-energy","display_name":"tidal stream energy","score":0.8029999732971191},{"id":"https://openalex.org/keywords/horizontal-axis-turbine","display_name":"horizontal axis turbine","score":0.8009999990463257}],"concepts":[{"id":"https://openalex.org/C132459708","wikidata":"https://www.wikidata.org/wiki/Q744069","display_name":"Extrapolation","level":2,"score":0.7494999766349792},{"id":"https://openalex.org/C48939323","wikidata":"https://www.wikidata.org/wiki/Q294879","display_name":"Wake","level":2,"score":0.6333000063896179},{"id":"https://openalex.org/C40018170","wikidata":"https://www.wikidata.org/wiki/Q184037","display_name":"Tidal power","level":2,"score":0.5498999953269958},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5178999900817871},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.48260000348091125},{"id":"https://openalex.org/C32230216","wikidata":"https://www.wikidata.org/wiki/Q7882499","display_name":"Uncertainty quantification","level":2,"score":0.4699999988079071},{"id":"https://openalex.org/C22019652","wikidata":"https://www.wikidata.org/wiki/Q331309","display_name":"Overfitting","level":3,"score":0.4537000060081482},{"id":"https://openalex.org/C2778449969","wikidata":"https://www.wikidata.org/wiki/Q130760","display_name":"Turbine","level":2,"score":0.4462999999523163},{"id":"https://openalex.org/C131675550","wikidata":"https://www.wikidata.org/wiki/Q7646884","display_name":"Surrogate model","level":2,"score":0.42480000853538513},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.39070001244544983},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.38609999418258667},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.3513999879360199},{"id":"https://openalex.org/C17020691","wikidata":"https://www.wikidata.org/wiki/Q139677","display_name":"Operator (biology)","level":5,"score":0.32820001244544983},{"id":"https://openalex.org/C177803969","wikidata":"https://www.wikidata.org/wiki/Q29205","display_name":"Uncertainty analysis","level":2,"score":0.32690000534057617},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.31690001487731934},{"id":"https://openalex.org/C38349280","wikidata":"https://www.wikidata.org/wiki/Q1434290","display_name":"Flow (mathematics)","level":2,"score":0.29580000042915344},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.2955999970436096},{"id":"https://openalex.org/C2776248883","wikidata":"https://www.wikidata.org/wiki/Q17154087","display_name":"Ocean observations","level":2,"score":0.28949999809265137},{"id":"https://openalex.org/C1633027","wikidata":"https://www.wikidata.org/wiki/Q815820","display_name":"Computational fluid dynamics","level":2,"score":0.27889999747276306},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2768999934196472},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.27239999175071716},{"id":"https://openalex.org/C177148314","wikidata":"https://www.wikidata.org/wiki/Q170084","display_name":"Generalization","level":2,"score":0.26080000400543213},{"id":"https://openalex.org/C163258240","wikidata":"https://www.wikidata.org/wiki/Q25342","display_name":"Power (physics)","level":2,"score":0.2605000138282776},{"id":"https://openalex.org/C78600449","wikidata":"https://www.wikidata.org/wiki/Q43302","display_name":"Wind power","level":2,"score":0.2538999915122986}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/math14081275","doi":"https://doi.org/10.3390/math14081275","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14081275","pdf_url":"https://www.mdpi.com/2227-7390/14/8/1275/pdf?version=1775917125","source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:fae0ee87dcd747f1954b4eb6e99f9f24","doi":null,"is_oa":true,"landing_page_url":"https://doaj.org/article/fae0ee87dcd747f1954b4eb6e99f9f24","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics, Vol 14, Iss 8, p 1275 (2026)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/math14081275","doi":"https://doi.org/10.3390/math14081275","is_oa":true,"landing_page_url":"https://doi.org/10.3390/math14081275","pdf_url":"https://www.mdpi.com/2227-7390/14/8/1275/pdf?version=1775917125","source":{"id":"https://openalex.org/S4210192031","display_name":"Mathematics","issn_l":"2227-7390","issn":["2227-7390"],"is_oa":true,"is_in_doaj":true,"is_core":true,"listed_in":["anvur-scientific","cwts-core","doaj","poland-20","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics","raw_type":"journal-article"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","display_name":"Affordable and clean energy","score":0.8485000133514404}],"sustainable_development_goals_aurora":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G6907103542","display_name":null,"funder_award_id":"52471310","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320321543","display_name":"China Postdoctoral Science Foundation","ror":"https://ror.org/0426zh255"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7154107343.pdf","grobid_xml":"https://content.openalex.org/works/W7154107343.grobid-xml"},"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Amid":[0],"the":[1,37,51,90,103,178,225],"global":[2],"transition":[3],"to":[4,18,152,161,169],"carbon":[5],"neutrality,":[6],"tidal":[7,41,120,235],"current":[8,236],"energy":[9,15,34,237],"has":[10],"become":[11],"a":[12,117,208,220],"strategic":[13],"sustainable":[14],"resource":[16],"due":[17],"its":[19],"high":[20],"predictability,":[21],"power":[22],"density,":[23],"and":[24,50,89,165,195,215,231],"environmental":[25],"compatibility.":[26],"Horizontal-axis":[27],"turbines":[28,42],"show":[29],"great":[30],"potential":[31],"for":[32,58,70,116,129,213],"marine":[33],"harvesting,":[35],"yet":[36],"large-scale":[38],"commercialization":[39],"of":[40,53,92,106,234],"is":[43,127,150],"severely":[44],"hindered":[45],"by":[46],"complex":[47],"wake":[48,107,217],"dynamics":[49],"lack":[52],"reliable,":[54],"efficient":[55,226],"prediction":[56,109,183,203,218],"tools":[57],"out-of-distribution":[59],"(OOD)":[60],"operating":[61],"conditions.":[62],"Traditional":[63],"high-fidelity":[64],"CFD":[65],"methods":[66],"are":[67],"computationally":[68],"prohibitive":[69],"engineering":[71,232],"optimization,":[72,230],"while":[73],"conventional":[74],"data-driven":[75,209],"surrogate":[76,210],"models":[77],"suffer":[78],"from":[79],"poor":[80],"extrapolation":[81,83,105,163],"performance,":[82],"collapse":[84,164],"near":[85],"training":[86],"parameter":[87],"boundaries,":[88],"absence":[91],"uncertainty":[93,172,198],"quantification.":[94],"To":[95],"address":[96],"these":[97],"bottlenecks,":[98],"this":[99],"study":[100],"focuses":[101],"on":[102,219],"OOD":[104,131,154],"flow":[108],"across":[110,133],"tip":[111],"speed":[112],"ratio":[113],"(TSR)":[114],"distributions":[115],"single":[118],"horizontal-axis":[119],"turbine.":[121],"A":[122,141],"CFD-generated":[123,222],"spatiotemporal":[124],"benchmark":[125],"dataset":[126],"constructed":[128],"comparative":[130],"evaluation":[132],"various":[134],"TSR":[135,187],"conditions":[136],"with":[137,201],"9504":[138],"total":[139],"samples.":[140],"novel":[142],"physics-constrained":[143],"Fourier":[144],"neural":[145],"operator":[146],"framework":[147],"named":[148],"TSR-FNO":[149],"proposed":[151,179],"improve":[153],"generalization.":[155],"The":[156],"model":[157],"integrates":[158],"TSR\u2013Lipschitz":[159],"regularization":[160],"suppress":[162],"Monte":[166],"Carlo":[167],"Dropout":[168],"provide":[170],"reliable":[171,216],"estimation.":[173],"Extensive":[174],"experiments":[175],"demonstrate":[176],"that":[177],"method":[180],"effectively":[181],"reduces":[182],"error":[184],"in":[185,192],"unseen":[186],"regimes,":[188],"mitigates":[189],"performance":[190],"degradation":[191],"far-field":[193],"extrapolation,":[194],"produces":[196],"well-calibrated":[197],"estimates":[199],"consistent":[200],"actual":[202],"confidence.":[204],"This":[205],"work":[206],"provides":[207],"modeling":[211],"strategy":[212],"fast":[214],"common":[221],"benchmark,":[223],"supporting":[224],"design,":[227],"array":[228],"layout":[229],"deployment":[233],"systems.":[238]},"counts_by_year":[{"year":2026,"cited_by_count":1}],"updated_date":"2026-10-14T16:39:01.820549","created_date":"2026-04-14T00:00:00"}