{"id":"https://openalex.org/W7125141203","doi":"https://doi.org/10.48550/arxiv.2601.13256","title":"Deep Neural networks for solving high-dimensional parabolic partial differential equations","display_name":"Deep Neural networks for solving high-dimensional parabolic partial differential equations","publication_year":2026,"publication_date":"2026-01-19","ids":{"openalex":"https://openalex.org/W7125141203","doi":"https://doi.org/10.48550/arxiv.2601.13256"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2601.13256","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.13256","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2601.13256","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5123509153","display_name":"Wenzhong Zhang","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Zhang, Wenzhong","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5123503348","display_name":"Zhenyuan Hu","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Hu, Zheyuan","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5123495149","display_name":"Wei Cai","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Cai, Wei","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5123496001","display_name":"George EM Karniadakis","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Karniadakis, George EM","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5123509153"],"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":0.9545000195503235,"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":0.9545000195503235,"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":0.008100000210106373,"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/T11416","display_name":"Numerical methods for differential equations","score":0.0052999998442828655,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/curse-of-dimensionality","display_name":"Curse of dimensionality","score":0.6614000201225281},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.631600022315979},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5519000291824341},{"id":"https://openalex.org/keywords/stochastic-partial-differential-equation","display_name":"Stochastic partial differential equation","score":0.4787999987602234},{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.4611000120639801},{"id":"https://openalex.org/keywords/benchmark","display_name":"Benchmark (surveying)","score":0.45899999141693115},{"id":"https://openalex.org/keywords/scalability","display_name":"Scalability","score":0.4253000020980835},{"id":"https://openalex.org/keywords/rendering","display_name":"Rendering (computer graphics)","score":0.40380001068115234},{"id":"https://openalex.org/keywords/differential-equation","display_name":"Differential equation","score":0.38600000739097595}],"concepts":[{"id":"https://openalex.org/C111030470","wikidata":"https://www.wikidata.org/wiki/Q1430460","display_name":"Curse of dimensionality","level":2,"score":0.6614000201225281},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.631600022315979},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5580000281333923},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5519000291824341},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5382000207901001},{"id":"https://openalex.org/C84629840","wikidata":"https://www.wikidata.org/wiki/Q16979017","display_name":"Stochastic partial differential equation","level":3,"score":0.4787999987602234},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.4611000120639801},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.45899999141693115},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.4253000020980835},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.412200003862381},{"id":"https://openalex.org/C205711294","wikidata":"https://www.wikidata.org/wiki/Q176953","display_name":"Rendering (computer graphics)","level":2,"score":0.40380001068115234},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.38839998841285706},{"id":"https://openalex.org/C78045399","wikidata":"https://www.wikidata.org/wiki/Q11214","display_name":"Differential equation","level":2,"score":0.38600000739097595},{"id":"https://openalex.org/C51955184","wikidata":"https://www.wikidata.org/wiki/Q1545585","display_name":"Stochastic differential equation","level":2,"score":0.37299999594688416},{"id":"https://openalex.org/C3019722297","wikidata":"https://www.wikidata.org/wiki/Q4440864","display_name":"High dimensional","level":2,"score":0.3481000065803528},{"id":"https://openalex.org/C70518039","wikidata":"https://www.wikidata.org/wiki/Q16000077","display_name":"Dimensionality reduction","level":2,"score":0.326200008392334},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.31349998712539673},{"id":"https://openalex.org/C2984842247","wikidata":"https://www.wikidata.org/wiki/Q197536","display_name":"Deep neural networks","level":3,"score":0.31060001254081726},{"id":"https://openalex.org/C97826883","wikidata":"https://www.wikidata.org/wiki/Q7069659","display_name":"Numerical partial differential equations","level":3,"score":0.29600000381469727},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.28519999980926514},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.2831000089645386},{"id":"https://openalex.org/C93226319","wikidata":"https://www.wikidata.org/wiki/Q193137","display_name":"Differential (mechanical device)","level":2,"score":0.27649998664855957},{"id":"https://openalex.org/C32230216","wikidata":"https://www.wikidata.org/wiki/Q7882499","display_name":"Uncertainty quantification","level":2,"score":0.26820001006126404},{"id":"https://openalex.org/C54067925","wikidata":"https://www.wikidata.org/wiki/Q3607962","display_name":"Elliptic partial differential equation","level":3,"score":0.2680000066757202},{"id":"https://openalex.org/C193386753","wikidata":"https://www.wikidata.org/wiki/Q1130396","display_name":"Approximations of \u03c0","level":2,"score":0.26739999651908875},{"id":"https://openalex.org/C2781067378","wikidata":"https://www.wikidata.org/wiki/Q17027399","display_name":"Interpretability","level":2,"score":0.257999986410141}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2601.13256","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.13256","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"article"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2601.13256","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2601.13256","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0,163],"numerical":[1],"solution":[2],"of":[3,16,45,52,121,160,169,180],"high":[4,65,93,124,181],"dimensional":[5,66,94,182],"partial":[6],"differential":[7,105],"equations":[8,148],"(PDEs)":[9],"is":[10],"severely":[11],"constrained":[12],"by":[13],"the":[14,43,77,118,131,155,161],"curse":[15],"dimensionality":[17],"(CoD),":[18],"rendering":[19],"classical":[20],"grid--based":[21],"methods":[22,62,98],"impractical":[23],"beyond":[24],"a":[25,37,57,167],"few":[26],"dimensions.":[27,53,125],"In":[28],"recent":[29],"years,":[30],"deep":[31],"neural":[32,89],"networks":[33,90],"have":[34],"emerged":[35],"as":[36],"promising":[38],"mesh":[39],"free":[40],"alternative,":[41],"enabling":[42],"approximation":[44],"PDE":[46,84],"solutions":[47],"in":[48,123,149],"tens":[49],"to":[50,60,116,151],"thousands":[51],"This":[54],"review":[55],"provides":[56],"tutorial--oriented":[58],"introduction":[59],"neural--network--based":[61],"for":[63,175],"solving":[64],"parabolic":[67],"PDEs,":[68],"emphasizing":[69],"conceptual":[70],"clarity":[71],"and":[72,91,102,108,137,140,146,158,172,177],"methodological":[73],"connections.":[74],"We":[75],"organize":[76],"literature":[78],"around":[79],"three":[80],"unifying":[81],"paradigms:":[82],"(i)":[83],"residual--based":[85],"approaches,":[86],"including":[87],"physicsinformed":[88],"their":[92],"variants;":[95],"(ii)":[96],"stochastic":[97,104],"derived":[99],"from":[100],"Feynman--Kac":[101],"backward":[103],"equation":[106],"formulations;":[107],"(iii)":[109],"hybrid":[110],"derivative--free":[111],"random":[112],"difference":[113],"approaches":[114],"designed":[115],"alleviate":[117],"computational":[119],"cost":[120],"derivatives":[122],"For":[126],"each":[127],"paradigm,":[128],"we":[129],"outline":[130],"underlying":[132],"mathematical":[133],"formulation,":[134],"algorithmic":[135],"implementation,":[136],"practical":[138],"strengths":[139],"limitations.":[141],"Representative":[142],"benchmark":[143],"problems--including":[144],"Hamilton--Jacobi--Bellman":[145],"Black--Scholes":[147],"up":[150],"1000":[152],"dimensions":[153],"--illustrate":[154],"scalability,":[156],"effectiveness,":[157],"accuracy":[159],"methods.":[162],"paper":[164],"concludes":[165],"with":[166],"discussion":[168],"open":[170],"challenges":[171],"future":[173],"directions":[174],"reliable":[176],"scalable":[178],"solvers":[179],"PDEs.":[183]},"counts_by_year":[],"updated_date":"2026-01-24T23:23:39.755997","created_date":"2026-01-22T00:00:00"}
