{"id":"https://openalex.org/W7225380593","doi":"https://doi.org/10.1063/5.0351160","title":"An explainable physics-informed neural network prototype for synthetic proppant transport in a two-dimensional fracture domain","display_name":"An explainable physics-informed neural network prototype for synthetic proppant transport in a two-dimensional fracture domain","publication_year":2026,"publication_date":"2026-11-01","ids":{"openalex":"https://openalex.org/W7225380593","doi":"https://doi.org/10.1063/5.0351160"},"language":"en","primary_location":{"id":"doi:10.1063/5.0351160","doi":"https://doi.org/10.1063/5.0351160","is_oa":false,"landing_page_url":"https://doi.org/10.1063/5.0351160","pdf_url":null,"source":{"id":"https://openalex.org/S22862804","display_name":"Physics of Fluids","issn_l":"1070-6631","issn":["1070-6631","1089-7666","1527-2435"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-d","jufo-1","ki-jl-1","norway-1","poland-100","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320257","host_organization_name":"American Institute of Physics","host_organization_lineage":["https://openalex.org/P4310320257"],"host_organization_lineage_names":["American Institute of Physics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Physics of Fluids","raw_type":"journal-article"},"volume":null,"type":"article","is_traditional":true,"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/A5060678293","display_name":"Moldir Zholdasbayeva","orcid":"https://orcid.org/0000-0002-6823-380X","observed_orcids":["https://orcid.org/0000-0002-6823-380X"]},"institutions":[{"id":"https://openalex.org/I60559429","display_name":"Nazarbayev University","ror":"https://ror.org/052bx8q98","country_code":"KZ","type":"education","lineage":["https://openalex.org/I60559429"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"Moldir Zholdasbayeva","raw_affiliation_strings":["School of Engineering, Department of Mining and Geosciences, Nazarbayev University 1 , Astana,"],"raw_orcid":"https://orcid.org/0000-0002-6823-380X","affiliations":[{"raw_affiliation_string":"School of Engineering, Department of Mining and Geosciences, Nazarbayev University 1 , Astana,","institution_ids":["https://openalex.org/I60559429"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5139086395","display_name":"Suleiman Hassan","orcid":"https://orcid.org/0009-0005-0671-0403","observed_orcids":["https://orcid.org/0009-0005-0671-0403"]},"institutions":[{"id":"https://openalex.org/I60559429","display_name":"Nazarbayev University","ror":"https://ror.org/052bx8q98","country_code":"KZ","type":"education","lineage":["https://openalex.org/I60559429"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"Suleiman Hassan","raw_affiliation_strings":["School of Engineering, Department of Mining and Geosciences, Nazarbayev University 1 , Astana,"],"raw_orcid":"https://orcid.org/0009-0005-0671-0403","affiliations":[{"raw_affiliation_string":"School of Engineering, Department of Mining and Geosciences, Nazarbayev University 1 , Astana,","institution_ids":["https://openalex.org/I60559429"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5031119717","display_name":"Dennis Delali Kwesi Wayo","orcid":"https://orcid.org/0000-0001-9980-6247","observed_orcids":["https://orcid.org/0000-0001-9980-6247"]},"institutions":[{"id":"https://openalex.org/I130701444","display_name":"Georgia Institute of Technology","ror":"https://ror.org/01zkghx44","country_code":"US","type":"education","lineage":["https://openalex.org/I130701444"]},{"id":"https://openalex.org/I60559429","display_name":"Nazarbayev University","ror":"https://ror.org/052bx8q98","country_code":"KZ","type":"education","lineage":["https://openalex.org/I60559429"]}],"countries":["KZ","US"],"is_corresponding":false,"raw_author_name":"Dennis Delali Kwesi Wayo","raw_affiliation_strings":["College of Computing, Georgia Institute of Technology 2 , Atlanta, Georgia 30332,","School of Engineering, Department of Mining and Geosciences, Nazarbayev University 1 , Astana,"],"raw_orcid":"https://orcid.org/0000-0001-9980-6247","affiliations":[{"raw_affiliation_string":"College of Computing, Georgia Institute of Technology 2 , Atlanta, Georgia 30332,","institution_ids":["https://openalex.org/I130701444"]},{"raw_affiliation_string":"School of Engineering, Department of Mining and Geosciences, Nazarbayev University 1 , Astana,","institution_ids":["https://openalex.org/I60559429"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079331414","display_name":"Sonny Irawan","orcid":"https://orcid.org/0000-0002-7444-868X","observed_orcids":["https://orcid.org/0000-0002-7444-868X"]},"institutions":[{"id":"https://openalex.org/I60559429","display_name":"Nazarbayev University","ror":"https://ror.org/052bx8q98","country_code":"KZ","type":"education","lineage":["https://openalex.org/I60559429"]}],"countries":["KZ"],"is_corresponding":true,"raw_author_name":"Sonny Irawan","raw_affiliation_strings":["School of Engineering, Department of Mining and Geosciences, Nazarbayev University 1 , Astana,"],"raw_orcid":"https://orcid.org/0000-0002-7444-868X","affiliations":[{"raw_affiliation_string":"School of Engineering, Department of Mining and Geosciences, Nazarbayev University 1 , Astana,","institution_ids":["https://openalex.org/I60559429"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5079331414"],"corresponding_institution_ids":["https://openalex.org/I60559429"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.0,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"38","issue":"11","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":0.9990000128746033,"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":0.9990000128746033,"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/T10635","display_name":"Hydraulic Fracturing and Reservoir Analysis","score":0.0003000000142492354,"subfield":{"id":"https://openalex.org/subfields/2103","display_name":"Fuel Technology"},"field":{"id":"https://openalex.org/fields/21","display_name":"Energy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11801","display_name":"Reservoir Engineering and Simulation Methods","score":9.999999747378752e-05,"subfield":{"id":"https://openalex.org/subfields/2103","display_name":"Fuel Technology"},"field":{"id":"https://openalex.org/fields/21","display_name":"Energy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/physics-informed-neural-networks","display_name":"physics-informed neural networks","score":0.9629999995231628},{"id":"https://openalex.org/keywords/proppant-transport","display_name":"proppant transport","score":0.972000002861023},{"id":"https://openalex.org/keywords/planar-fractures","display_name":"planar fractures","score":0.5419999957084656},{"id":"https://openalex.org/keywords/convection-diffusion","display_name":"advection-diffusion","score":0.6970000267028809},{"id":"https://openalex.org/keywords/proppant-concentration","display_name":"proppant concentration","score":0.6830000281333923},{"id":"https://openalex.org/keywords/explainable-ai","display_name":"explainable AI","score":0.6919999718666077},{"id":"https://openalex.org/keywords/shap","display_name":"SHAP","score":0.9210000038146973},{"id":"https://openalex.org/keywords/integrated-gradients","display_name":"integrated gradients","score":0.9620000123977661},{"id":"https://openalex.org/keywords/residual-maps","display_name":"residual maps","score":0.5360000133514404},{"id":"https://openalex.org/keywords/synthetic-benchmark","display_name":"synthetic benchmark","score":0.7300000190734863},{"id":"https://openalex.org/keywords/velocity-divergence","display_name":"velocity divergence","score":0.3190000057220459},{"id":"https://openalex.org/keywords/pressure-gradient","display_name":"pressure gradient","score":0.460999995470047},{"id":"https://openalex.org/keywords/fracture-flow","display_name":"fracture flow","score":0.4320000112056732}],"concepts":[{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.722000002861023},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6486999988555908},{"id":"https://openalex.org/C185798385","wikidata":"https://www.wikidata.org/wiki/Q1161707","display_name":"Benchmark (surveying)","level":2,"score":0.6201000213623047},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.5877000093460083},{"id":"https://openalex.org/C122383733","wikidata":"https://www.wikidata.org/wiki/Q865920","display_name":"Approximation error","level":2,"score":0.5824999809265137},{"id":"https://openalex.org/C139945424","wikidata":"https://www.wikidata.org/wiki/Q1940696","display_name":"Mean squared error","level":2,"score":0.5806000232696533},{"id":"https://openalex.org/C57691317","wikidata":"https://www.wikidata.org/wiki/Q1289248","display_name":"Scalar (mathematics)","level":2,"score":0.49799999594688416},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.47279998660087585},{"id":"https://openalex.org/C43369102","wikidata":"https://www.wikidata.org/wiki/Q2307625","display_name":"Fracture (geology)","level":2,"score":0.47029998898506165},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.4537000060081482},{"id":"https://openalex.org/C134786449","wikidata":"https://www.wikidata.org/wiki/Q3391255","display_name":"Planar","level":2,"score":0.4226999878883362},{"id":"https://openalex.org/C71907059","wikidata":"https://www.wikidata.org/wiki/Q223323","display_name":"Root mean square","level":2,"score":0.40230000019073486},{"id":"https://openalex.org/C155032097","wikidata":"https://www.wikidata.org/wiki/Q798503","display_name":"Backpropagation","level":3,"score":0.3928000032901764},{"id":"https://openalex.org/C60718061","wikidata":"https://www.wikidata.org/wiki/Q1414747","display_name":"Momentum (technical analysis)","level":2,"score":0.34200000762939453},{"id":"https://openalex.org/C11577676","wikidata":"https://www.wikidata.org/wiki/Q134237","display_name":"Square root","level":2,"score":0.32409998774528503},{"id":"https://openalex.org/C188154048","wikidata":"https://www.wikidata.org/wiki/Q6803609","display_name":"Mean absolute error","level":3,"score":0.314300000667572},{"id":"https://openalex.org/C103824480","wikidata":"https://www.wikidata.org/wiki/Q185889","display_name":"Time domain","level":2,"score":0.27320000529289246},{"id":"https://openalex.org/C135692309","wikidata":"https://www.wikidata.org/wiki/Q111124","display_name":"Square (algebra)","level":2,"score":0.272599995136261},{"id":"https://openalex.org/C110521144","wikidata":"https://www.wikidata.org/wiki/Q193460","display_name":"Scalar field","level":2,"score":0.27230000495910645},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.2711000144481659},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2547999918460846}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1063/5.0351160","doi":"https://doi.org/10.1063/5.0351160","is_oa":false,"landing_page_url":"https://doi.org/10.1063/5.0351160","pdf_url":null,"source":{"id":"https://openalex.org/S22862804","display_name":"Physics of Fluids","issn_l":"1070-6631","issn":["1070-6631","1089-7666","1527-2435"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["cwts-core","dongbi-d","jufo-1","ki-jl-1","norway-1","poland-100","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320257","host_organization_name":"American Institute of Physics","host_organization_lineage":["https://openalex.org/P4310320257"],"host_organization_lineage_names":["American Institute of Physics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Physics of Fluids","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[{"id":"https://openalex.org/F4320327695","display_name":"Nazarbayev University","ror":"https://ror.org/052bx8q98"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":17,"referenced_works":["https://openalex.org/W1984343068","https://openalex.org/W2001263565","https://openalex.org/W2025319708","https://openalex.org/W2027431847","https://openalex.org/W2169393322","https://openalex.org/W2765376001","https://openalex.org/W2807263755","https://openalex.org/W2899283552","https://openalex.org/W3017912079","https://openalex.org/W3144994396","https://openalex.org/W3163993681","https://openalex.org/W3204698378","https://openalex.org/W3205073660","https://openalex.org/W3210722409","https://openalex.org/W4288039037","https://openalex.org/W4408336071","https://openalex.org/W4412981151"],"related_works":["https://openalex.org/W4415777164","https://openalex.org/W7169876884","https://openalex.org/W4406727617","https://openalex.org/W4398616389","https://openalex.org/W4408186437","https://openalex.org/W7196963401","https://openalex.org/W7140189354","https://openalex.org/W2955939954","https://openalex.org/W3188026634","https://openalex.org/W4387881554"],"abstract_inverted_index":{"This":[0],"study":[1],"examines":[2],"a":[3,11,17,36,79],"single-domain":[4],"physics-informed":[5],"neural":[6],"network":[7],"(PINN)":[8],"prototype":[9],"on":[10,84],"normalized":[12],"two-dimensional":[13],"rectangular":[14],"domain":[15],"representing":[16],"planar":[18],"fracture":[19,194],"cross":[20],"section.":[21],"The":[22,54,151],"recovered":[23],"implementation":[24],"maps":[25],"(x,y,t)":[26],"to":[27],"(u,v,p,Cp)":[28],"and":[29,35,66,68,90,99,132,142,148,160,171,180],"penalizes":[30],"velocity":[31],"divergence,":[32],"pressure":[33],"gradients,":[34],"scalar":[37],"advection\u2013diffusion\u2013settling":[38],"residual":[39,153,178],"together":[40],"with":[41,93],"supervised":[42],"concentration":[43,61],"error;":[44],"it":[45],"does":[46],"not":[47],"enforce":[48],"the":[49,103,109,113,164,181],"full":[50,152],"Navier\u2013Stokes":[51],"momentum":[52],"equations.":[53],"archived":[55],"run":[56],"uses":[57],"all":[58],"5000":[59],"generated":[60],"labels":[62],"for":[63,144,169],"both":[64],"training":[65],"evaluation":[67],"reports":[69],"an":[70,185],"in-sample":[71],"R2=0.635.":[72],"To":[73],"provide":[74,184],"independent":[75],"verification,":[76],"we":[77],"add":[78],"seeded":[80],"3500/750/750":[81],"training/validation/test":[82],"protocol":[83],"manufactured":[85],"analytical":[86],"u,":[87,145],"v,":[88,146],"p,":[89,147],"Cp":[91,117],"fields":[92],"exact":[94],"forcing.":[95],"Matched":[96],"data-only,":[97],"reported-proxy,":[98],"full-residual":[100,114],"PINNs":[101],"use":[102],"same":[104],"corrected":[105],"3\u201364\u201364\u201364\u20134":[106],"architecture.":[107],"On":[108],"750":[110],"held-out":[111,155],"points,":[112],"PINN":[115],"gives":[116],"root":[118],"mean":[119,123],"square":[120],"error":[121,126,130],"1.71\u00d710\u22124,":[122],"absolute":[124,129],"percentage":[125],"0.610%,":[127],"maximum":[128],"9.12\u00d710\u22124,":[131],"R2=0.973.":[133],"Its":[134],"relative":[135],"L2":[136],"errors":[137],"are":[138],"0.0228,":[139],"0.606,":[140],"0.356,":[141],"0.00834":[143],"Cp,":[149],"respectively.":[150],"improves":[154],"concentration,":[156],"horizontal":[157],"velocity,":[158],"continuity,":[159],"scalar-transport":[161],"consistency,":[162],"whereas":[163],"data-only":[165],"model":[166],"remains":[167],"better":[168],"v":[170],"p.":[172],"SHapley":[173],"Additive":[174],"exPlanations,":[175],"Integrated":[176],"Gradients,":[177],"maps,":[179],"controlled":[182],"comparison":[183],"inspectable":[186],"synthetic":[187],"benchmark":[188],"without":[189],"implying":[190],"experimental":[191],"or":[192],"high-fidelity":[193],"validation.":[195]},"counts_by_year":[],"updated_date":"2026-11-08T12:12:28.789561","created_date":"2026-11-07T00:00:00"}
