{"id":"https://openalex.org/W3011147100","doi":"https://doi.org/10.1137/19m1260141","title":"Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks","display_name":"Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks","publication_year":2020,"publication_date":"2020-01-01","ids":{"openalex":"https://openalex.org/W3011147100","doi":"https://doi.org/10.1137/19m1260141","mag":"3011147100"},"language":"en","primary_location":{"id":"doi:10.1137/19m1260141","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1260141","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://www.osti.gov/servlets/purl/2281784","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5035174542","display_name":"Dongkun Zhang","orcid":"https://orcid.org/0000-0002-0811-4964"},"institutions":[{"id":"https://openalex.org/I27804330","display_name":"Brown University","ror":"https://ror.org/05gq02987","country_code":"US","type":"education","lineage":["https://openalex.org/I27804330"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Dongkun Zhang","raw_affiliation_strings":["Brown University, Providence, RI (United States)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Brown University, Providence, RI (United States)","institution_ids":["https://openalex.org/I27804330"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5108670524","display_name":"Ling Guo","orcid":null},"institutions":[{"id":"https://openalex.org/I21945476","display_name":"Shanghai Normal University","ror":"https://ror.org/01cxqmw89","country_code":"CN","type":"education","lineage":["https://openalex.org/I21945476"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ling Guo","raw_affiliation_strings":["Shanghai Normal University (China)"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Shanghai Normal University (China)","institution_ids":["https://openalex.org/I21945476"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5009658255","display_name":"George Em Karniadakis","orcid":"https://orcid.org/0000-0002-9713-7120"},"institutions":[{"id":"https://openalex.org/I27804330","display_name":"Brown University","ror":"https://ror.org/05gq02987","country_code":"US","type":"education","lineage":["https://openalex.org/I27804330"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"George Em Karniadakis","raw_affiliation_strings":["Brown University, Providence, RI (United States)"],"raw_orcid":"https://orcid.org/0000-0002-9713-7120","affiliations":[{"raw_affiliation_string":"Brown University, Providence, RI (United States)","institution_ids":["https://openalex.org/I27804330"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5035174542"],"corresponding_institution_ids":["https://openalex.org/I27804330"],"apc_list":null,"apc_paid":null,"fwci":20.0017,"has_fulltext":true,"cited_by_count":287,"citation_normalized_percentile":{"value":0.99797661,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":96,"max":100},"biblio":{"volume":"42","issue":"2","first_page":"A639","last_page":"A665"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9998999834060669,"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":0.9998999834060669,"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.9847999811172485,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6335490942001343},{"id":"https://openalex.org/keywords/stochastic-partial-differential-equation","display_name":"Stochastic partial differential equation","score":0.5743194818496704},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5644253492355347},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5415412187576294},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5386573672294617},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.5126048922538757},{"id":"https://openalex.org/keywords/stochastic-differential-equation","display_name":"Stochastic differential equation","score":0.47850504517555237},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.45030295848846436},{"id":"https://openalex.org/keywords/stochastic-neural-network","display_name":"Stochastic neural network","score":0.438546359539032},{"id":"https://openalex.org/keywords/ordinary-differential-equation","display_name":"Ordinary differential equation","score":0.43850457668304443},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.3650887608528137},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.31403741240501404},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.2795192003250122},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20739635825157166},{"id":"https://openalex.org/keywords/recurrent-neural-network","display_name":"Recurrent neural network","score":0.19245797395706177},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.11712047457695007}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6335490942001343},{"id":"https://openalex.org/C84629840","wikidata":"https://www.wikidata.org/wiki/Q16979017","display_name":"Stochastic partial differential equation","level":3,"score":0.5743194818496704},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5644253492355347},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5415412187576294},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5386573672294617},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.5126048922538757},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.47850504517555237},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.45030295848846436},{"id":"https://openalex.org/C86582703","wikidata":"https://www.wikidata.org/wiki/Q7617824","display_name":"Stochastic neural network","level":4,"score":0.438546359539032},{"id":"https://openalex.org/C51544822","wikidata":"https://www.wikidata.org/wiki/Q465274","display_name":"Ordinary differential equation","level":3,"score":0.43850457668304443},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.3650887608528137},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.31403741240501404},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.2795192003250122},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20739635825157166},{"id":"https://openalex.org/C147168706","wikidata":"https://www.wikidata.org/wiki/Q1457734","display_name":"Recurrent neural network","level":3,"score":0.19245797395706177},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.11712047457695007},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/19m1260141","is_oa":false,"landing_page_url":"https://doi.org/10.1137/19m1260141","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},{"id":"pmh:oai:osti.gov:2281784","is_oa":true,"landing_page_url":"https://www.osti.gov/biblio/2281784","pdf_url":"https://www.osti.gov/servlets/purl/2281784","source":{"id":"https://openalex.org/S4306402487","display_name":"OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I139351228","host_organization_name":"Office of Scientific and Technical Information","host_organization_lineage":["https://openalex.org/I139351228"],"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":null}],"best_oa_location":{"id":"pmh:oai:osti.gov:2281784","is_oa":true,"landing_page_url":"https://www.osti.gov/biblio/2281784","pdf_url":"https://www.osti.gov/servlets/purl/2281784","source":{"id":"https://openalex.org/S4306402487","display_name":"OSTI OAI (U.S. Department of Energy Office of Scientific and Technical Information)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I139351228","host_organization_name":"Office of Scientific and Technical Information","host_organization_lineage":["https://openalex.org/I139351228"],"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":null},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G2403815124","display_name":null,"funder_award_id":"11671265","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3353826364","display_name":null,"funder_award_id":"W911NF-12-2-0023","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G7620038843","display_name":null,"funder_award_id":"100028801-BROWN-APP","funder_id":"https://openalex.org/F4320306084","funder_display_name":"U.S. Department of Energy"},{"id":"https://openalex.org/G8512854826","display_name":null,"funder_award_id":"TZ2018001","funder_id":"https://openalex.org/F4320336025","funder_display_name":"Science Challenge Project"},{"id":"https://openalex.org/G8536560856","display_name":null,"funder_award_id":"TZ2018001","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320306084","display_name":"U.S. Department of Energy","ror":"https://ror.org/01bj3aw27"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320336025","display_name":"Science Challenge Project","ror":null}],"has_content":{"grobid_xml":false,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3011147100.pdf"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W1972098742","https://openalex.org/W1973359087","https://openalex.org/W2018159038","https://openalex.org/W2022264564","https://openalex.org/W2027491117","https://openalex.org/W2030308624","https://openalex.org/W2060235470","https://openalex.org/W2077180405","https://openalex.org/W2087257783","https://openalex.org/W2095354302","https://openalex.org/W2103496339","https://openalex.org/W2119334003","https://openalex.org/W2139923370","https://openalex.org/W2143591652","https://openalex.org/W2148666448","https://openalex.org/W2158985775","https://openalex.org/W2267184893","https://openalex.org/W2611636794","https://openalex.org/W2745110207","https://openalex.org/W2889522259","https://openalex.org/W2890968382","https://openalex.org/W2963843310","https://openalex.org/W3105648287"],"related_works":["https://openalex.org/W620239920","https://openalex.org/W109055908","https://openalex.org/W2999616507","https://openalex.org/W2060658157","https://openalex.org/W2734530159","https://openalex.org/W1546584948","https://openalex.org/W2972896374","https://openalex.org/W2164842548","https://openalex.org/W1762206374","https://openalex.org/W2085400897"],"abstract_inverted_index":{"One":[0],"of":[1,12,31,64,70,110,118,126,137,140,157,175,214,256,266,272,319,352,363],"the":[2,9,39,58,89,98,102,107,111,123,127,131,138,141,151,154,158,165,173,181,186,199,212,215,238,242,257,263,270,273,284,289,302,306,317,320,337,360],"open":[3],"problems":[4,197,278],"in":[5,34,164,180,250,323,348],"scientific":[6,35],"computing":[7],"is":[8,161],"long-time":[10],"integration":[11,255],"nonlinear":[13],"stochastic":[14,50,71,195,226,258],"partial":[15],"differential":[16,73],"equations":[17],"(SPDEs),":[18],"especially":[19,269],"with":[20,106,113,198,229,237,279],"arbitrary":[21],"initial":[22,231,312],"data.":[23],"We":[24,210],"address":[25,359],"this":[26,251],"problem":[27,304],"by":[28],"taking":[29],"advantage":[30],"recent":[32],"advances":[33],"machine":[36],"learning":[37],"and":[38,44,67,169,202,305,326],"spectral":[40],"dynamically":[41],"orthogonal":[42],"(DO)":[43],"borthogonal":[45],"(BO)":[46],"methods":[47,56,96,133,188,217,245,322,339],"for":[48,84,122,207,276,365],"representing":[49],"processes.":[51],"The":[52,94],"recently":[53],"introduced":[54],"DO/BO":[55,99,128,143,244],"reduce":[57],"SPDE":[59],"to":[60,192,315,343],"solving":[61,85],"a":[62,68,349],"system":[63,69],"deterministic":[65,230],"PDEs":[66],"ordinary":[72],"equations.":[74],"Specifically,":[75],"we":[76,147,170,233,261,299],"propose":[77],"two":[78],"new":[79],"physics-informed":[80],"neural":[81,90],"networks":[82],"(PINNs)":[83],"time-dependent":[86,194],"SPDEs,":[87],"namely":[88],"network":[91],"(NN)-DO/BO":[92],"methods.":[93,144],"proposed":[95,216,239],"incorporate":[97],"constraints":[100],"into":[101],"loss":[103],"function":[104],"(along":[105],"modal":[108],"decomposition":[109],"SPDE)":[112],"an":[114],"implicit":[115],"form":[116],"instead":[117],"generating":[119],"explicit":[120],"expressions":[121],"temporal":[124],"derivatives":[125],"modes.":[129],"Hence,":[130],"NN-DO/BO":[132,187,267,321,338],"can":[134,171,189,340],"overcome":[135],"some":[136],"drawbacks":[139],"original":[142,166,182,243,290],"For":[145],"example,":[146],"do":[148],"not":[149],"need":[150],"assumption":[152,174],"that":[153,336],"covariance":[155],"matrix":[156],"random":[159],"coefficients":[160],"invertible":[162],"as":[163,179,206],"DO":[167],"method,":[168],"remove":[172],"no":[176],"eigenvalue":[177,281],"crossing":[178],"BO":[183,291],"method.":[184],"Moreover,":[185],"be":[190,247,341],"used":[191],"solve":[193],"inverse":[196,307,325],"same":[200,203],"formulation":[201],"computational":[204],"complexity":[205],"forward":[208,303,366],"problems.":[209,329,367],"demonstrate":[211,335],"capability":[213],"via":[218],"several":[219],"numerical":[220],"examples,":[221],"namely:":[222],"(1)":[223],"A":[224],"linear":[225],"advection":[227],"equation":[228],"condition:":[232],"obtain":[234],"good":[235,264],"results":[236,334],"methods,":[240,268],"while":[241,288],"cannot":[246],"applied":[248],"directly":[249],"case.":[252],"(2)":[253],"Long-time":[254],"Burgers'":[259],"equation:":[260,298],"show":[262],"performance":[265],"effectiveness":[271],"NN-BO":[274],"approach":[275],"such":[277],"many":[280],"crossings":[282],"during":[283],"whole":[285],"time":[286],"evolution,":[287],"method":[292],"fails.":[293],"(3)":[294],"Nonlinear":[295],"reaction":[296],"diffusion":[297],"consider":[300],"both":[301],"problems,":[308,354],"including":[309],"very":[310],"noisy":[311],"point":[313],"values,":[314],"investigate":[316],"flexibility":[318],"handling":[324],"mixed":[327],"type":[328],"Taken":[330],"together,":[331],"these":[332],"simulation":[333],"employed":[342],"effectively":[344],"quantify":[345],"uncertainty":[346],"propagation":[347],"wide":[350],"range":[351],"physical":[353],"but":[355],"future":[356],"work":[357],"should":[358],"efficiency":[361],"issue":[362],"PINNs":[364]},"counts_by_year":[{"year":2026,"cited_by_count":30},{"year":2025,"cited_by_count":54},{"year":2024,"cited_by_count":50},{"year":2023,"cited_by_count":52},{"year":2022,"cited_by_count":46},{"year":2021,"cited_by_count":35},{"year":2020,"cited_by_count":17},{"year":2019,"cited_by_count":3}],"updated_date":"2026-05-08T15:41:06.802602","created_date":"2025-10-10T00:00:00"}
