{"id":"https://openalex.org/W7158673421","doi":"https://doi.org/10.48550/arxiv.2604.25943","title":"A Randomized PDE Energy driven Iterative Framework for Efficient and Stable PDE Solutions","display_name":"A Randomized PDE Energy driven Iterative Framework for Efficient and Stable PDE Solutions","publication_year":2026,"publication_date":"2026-04-17","ids":{"openalex":"https://openalex.org/W7158673421","doi":"https://doi.org/10.48550/arxiv.2604.25943"},"language":null,"primary_location":{"id":"doi:10.48550/arxiv.2604.25943","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.25943","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"type":"preprint","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.48550/arxiv.2604.25943","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5132850266","display_name":"Yi Bing","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Bing, Yi","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5134914152","display_name":"Zheng Ran","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Ran, Zheng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5132899032","display_name":"Fu Jinyang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Jinyang, Fu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5134927636","display_name":"Liu Long","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Long, Liu","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]},{"author_position":"last","author":{"id":"https://openalex.org/A5134903812","display_name":"Peng Xiang","orcid":null},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Xiang, Peng","raw_affiliation_strings":[],"raw_orcid":null,"affiliations":[]}],"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":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":0.7960000038146973,"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":0.7960000038146973,"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/T10514","display_name":"Numerical methods in engineering","score":0.04100000113248825,"subfield":{"id":"https://openalex.org/subfields/2211","display_name":"Mechanics of Materials"},"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/T11948","display_name":"Machine Learning in Materials Science","score":0.023000000044703484,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.6331999897956848},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5853000283241272},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5591999888420105},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.48820000886917114},{"id":"https://openalex.org/keywords/matrix","display_name":"Matrix (chemical analysis)","score":0.4505999982357025},{"id":"https://openalex.org/keywords/boundary-value-problem","display_name":"Boundary value problem","score":0.3752000033855438},{"id":"https://openalex.org/keywords/boundary","display_name":"Boundary (topology)","score":0.37400001287460327},{"id":"https://openalex.org/keywords/energy","display_name":"Energy (signal processing)","score":0.37130001187324524}],"concepts":[{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.6331999897956848},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5853000283241272},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5591999888420105},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5336999893188477},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.5091000199317932},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.48820000886917114},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.4505999982357025},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.44780001044273376},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4327999949455261},{"id":"https://openalex.org/C182310444","wikidata":"https://www.wikidata.org/wiki/Q1332643","display_name":"Boundary value problem","level":2,"score":0.3752000033855438},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.37400001287460327},{"id":"https://openalex.org/C186370098","wikidata":"https://www.wikidata.org/wiki/Q442787","display_name":"Energy (signal processing)","level":2,"score":0.37130001187324524},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.3621000051498413},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.3564999997615814},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.35580000281333923},{"id":"https://openalex.org/C204323151","wikidata":"https://www.wikidata.org/wiki/Q905424","display_name":"Range (aeronautics)","level":2,"score":0.34929999709129333},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.3248000144958496},{"id":"https://openalex.org/C48044578","wikidata":"https://www.wikidata.org/wiki/Q727490","display_name":"Scalability","level":2,"score":0.29760000109672546},{"id":"https://openalex.org/C129747778","wikidata":"https://www.wikidata.org/wiki/Q1014918","display_name":"Burgers' equation","level":3,"score":0.28679999709129333},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.28619998693466187},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.28529998660087585},{"id":"https://openalex.org/C32230216","wikidata":"https://www.wikidata.org/wiki/Q7882499","display_name":"Uncertainty quantification","level":2,"score":0.2761000096797943},{"id":"https://openalex.org/C2780799671","wikidata":"https://www.wikidata.org/wiki/Q17087362","display_name":"Transient (computer programming)","level":2,"score":0.2687999904155731},{"id":"https://openalex.org/C186867907","wikidata":"https://www.wikidata.org/wiki/Q1575416","display_name":"Parabolic partial differential equation","level":3,"score":0.26669999957084656},{"id":"https://openalex.org/C176321772","wikidata":"https://www.wikidata.org/wiki/Q1430640","display_name":"Numerical stability","level":3,"score":0.2623000144958496}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.48550/arxiv.2604.25943","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.25943","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":null,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"doi:10.48550/arxiv.2604.25943","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2604.25943","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":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/7","score":0.7651068568229675,"display_name":"Affordable and clean energy"}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"Efficient":[0],"and":[1,13,33,109,117,139,164,174,193],"stable":[2,123],"solution":[3,129],"of":[4,136,149],"partial":[5],"differential":[6],"equations":[7],"(PDEs)":[8],"is":[9,101],"central":[10],"to":[11,103,125,178],"scientific":[12],"engineering":[14,194],"applications,":[15],"yet":[16],"existing":[17],"numerical":[18,180],"solvers":[19],"rely":[20],"heavily":[21],"on":[22,59],"matrix":[23,61],"based":[24,28,62],"discretizations,":[25],"while":[26,90],"learning":[27],"methods":[29],"require":[30],"costly":[31],"training":[32],"often":[34],"suffer":[35],"from":[36,130],"limited":[37],"generalization.":[38],"In":[39],"this":[40],"work,":[41],"we":[42],"proposes":[43],"a":[44,146,171,183],"PDE":[45,81,188],"energy":[46,82],"driven":[47,68,83],"framework":[48,160,169],"that":[49,158],"solves":[50],"PDEs":[51],"through":[52,80],"physically":[53,175],"constrained":[54],"diffusion":[55],"iterations,":[56],"without":[57],"relying":[58],"classical":[60],"finite":[63],"element":[64],"assembly":[65],"or":[66],"data":[67],"neural":[69],"network":[70],"training.":[71],"The":[72,98],"proposed":[73,99,168],"method":[74],"evolves":[75],"arbitrary":[76],"random":[77,131],"initial":[78],"fields":[79],"implicit":[84],"iterations":[85],"combined":[86],"with":[87,133,154],"Gaussian":[88],"smoothing,":[89],"strictly":[91],"enforcing":[92],"boundary":[93],"conditions":[94],"at":[95],"each":[96],"iteration.":[97],"formulation":[100],"applied":[102],"representative":[104],"one":[105],"dimensional":[106],"Poisson,":[107],"Heat,":[108],"viscous":[110],"Burgers":[111],"equations,":[112],"covering":[113],"both":[114,191],"steady":[115],"state":[116],"transient":[118],"problems.":[119],"Numerical":[120],"results":[121],"demonstrate":[122],"convergence":[124],"the":[126,159,167],"unique":[127],"physical":[128],"initializations,":[132],"accurate":[134],"resolution":[135],"sharp":[137],"gradients":[138],"controlled":[140],"Mean":[141],"Squared":[142],"Error":[143],"(MSE)":[144],"across":[145],"wide":[147],"range":[148],"discretization":[150],"parameters.":[151],"Detailed":[152],"comparisons":[153],"analytical":[155],"solutions":[156,189],"indicate":[157],"achieves":[161],"competitive":[162],"accuracy":[163],"stability.":[165],"Overall,":[166],"provides":[170],"fast,":[172],"flexible,":[173],"consistent":[176],"alternative":[177],"traditional":[179],"solvers,":[181],"offering":[182],"potential":[184],"pathway":[185],"for":[186],"scalable":[187],"in":[190],"research":[192],"applications.":[195]},"counts_by_year":[],"updated_date":"2026-07-01T06:00:48.157686","created_date":"2026-05-01T00:00:00"}