{"id":"https://openalex.org/W4411002368","doi":"https://doi.org/10.3390/a18060334","title":"Adaptive Grid Generation by Solving One-Dimensional Diffusion Equation Using Physics-Informed Neural Networks","display_name":"Adaptive Grid Generation by Solving One-Dimensional Diffusion Equation Using Physics-Informed Neural Networks","publication_year":2025,"publication_date":"2025-06-03","ids":{"openalex":"https://openalex.org/W4411002368","doi":"https://doi.org/10.3390/a18060334"},"language":"en","primary_location":{"id":"doi:10.3390/a18060334","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a18060334","pdf_url":"https://www.mdpi.com/1999-4893/18/6/334/pdf?version=1748931663","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"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":"Algorithms","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1999-4893/18/6/334/pdf?version=1748931663","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5069858823","display_name":"Olzhas Turar","orcid":"https://orcid.org/0000-0002-6720-0045"},"institutions":[{"id":"https://openalex.org/I185571130","display_name":"Al-Farabi Kazakh National University","ror":"https://ror.org/03q0vrn42","country_code":"KZ","type":"education","lineage":["https://openalex.org/I185571130"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"Olzhas Turar","raw_affiliation_strings":["Faculty of Information Technologies, Al-Farabi Kazakh National University, Almaty 050000, Kazakhstan"],"affiliations":[{"raw_affiliation_string":"Faculty of Information Technologies, Al-Farabi Kazakh National University, Almaty 050000, Kazakhstan","institution_ids":["https://openalex.org/I185571130"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020480885","display_name":"Maksat Mustafin","orcid":"https://orcid.org/0000-0002-3655-0771"},"institutions":[{"id":"https://openalex.org/I185571130","display_name":"Al-Farabi Kazakh National University","ror":"https://ror.org/03q0vrn42","country_code":"KZ","type":"education","lineage":["https://openalex.org/I185571130"]}],"countries":["KZ"],"is_corresponding":true,"raw_author_name":"Maksat Mustafin","raw_affiliation_strings":["Faculty of Information Technologies, Al-Farabi Kazakh National University, Almaty 050000, Kazakhstan"],"affiliations":[{"raw_affiliation_string":"Faculty of Information Technologies, Al-Farabi Kazakh National University, Almaty 050000, Kazakhstan","institution_ids":["https://openalex.org/I185571130"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5046141722","display_name":"Darkhan Akhmed-Zaki","orcid":"https://orcid.org/0000-0001-8100-8263"},"institutions":[{"id":"https://openalex.org/I4210151108","display_name":"M.Auezov South Kazakhstan State University","ror":"https://ror.org/05xzaq362","country_code":"KZ","type":"education","lineage":["https://openalex.org/I4210151108"]}],"countries":["KZ"],"is_corresponding":false,"raw_author_name":"Darkhan Akhmed-Zaki","raw_affiliation_strings":["Mukhtar Auezov South Kazakhstan Research University, Shymkent 160000, Kazakhstan"],"affiliations":[{"raw_affiliation_string":"Mukhtar Auezov South Kazakhstan Research University, Shymkent 160000, Kazakhstan","institution_ids":["https://openalex.org/I4210151108"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5020480885"],"corresponding_institution_ids":["https://openalex.org/I185571130"],"apc_list":{"value":1400,"currency":"CHF","value_usd":1515},"apc_paid":{"value":1400,"currency":"CHF","value_usd":1515},"fwci":0.0,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":{"value":0.10458957,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":"18","issue":"6","first_page":"334","last_page":"334"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9998000264167786,"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":0.9998000264167786,"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/T10339","display_name":"Advanced Numerical Methods in Computational Mathematics","score":0.9869999885559082,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/T10360","display_name":"Fluid Dynamics and Turbulent Flows","score":0.9839000105857849,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"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/artificial-neural-network","display_name":"Artificial neural network","score":0.6757570505142212},{"id":"https://openalex.org/keywords/grid","display_name":"Grid","score":0.6740416884422302},{"id":"https://openalex.org/keywords/diffusion","display_name":"Diffusion","score":0.5376364588737488},{"id":"https://openalex.org/keywords/diffusion-equation","display_name":"Diffusion equation","score":0.5133318305015564},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.5030483603477478},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.428294837474823},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.3747131824493408},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.2728920578956604},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2721182703971863},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.2635098099708557},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.11264243721961975},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.08575844764709473}],"concepts":[{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.6757570505142212},{"id":"https://openalex.org/C187691185","wikidata":"https://www.wikidata.org/wiki/Q2020720","display_name":"Grid","level":2,"score":0.6740416884422302},{"id":"https://openalex.org/C69357855","wikidata":"https://www.wikidata.org/wiki/Q163214","display_name":"Diffusion","level":2,"score":0.5376364588737488},{"id":"https://openalex.org/C571446","wikidata":"https://www.wikidata.org/wiki/Q6510183","display_name":"Diffusion equation","level":3,"score":0.5133318305015564},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.5030483603477478},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.428294837474823},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.3747131824493408},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2728920578956604},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2721182703971863},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.2635098099708557},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.11264243721961975},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.08575844764709473},{"id":"https://openalex.org/C2780378061","wikidata":"https://www.wikidata.org/wiki/Q25351891","display_name":"Service (business)","level":2,"score":0.0},{"id":"https://openalex.org/C136264566","wikidata":"https://www.wikidata.org/wiki/Q159810","display_name":"Economy","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.3390/a18060334","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a18060334","pdf_url":"https://www.mdpi.com/1999-4893/18/6/334/pdf?version=1748931663","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"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":"Algorithms","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:d34882d2305e407b92a67bbc6d928bc8","is_oa":true,"landing_page_url":"https://doaj.org/article/d34882d2305e407b92a67bbc6d928bc8","pdf_url":null,"source":{"id":"https://openalex.org/S112646816","display_name":"SHILAP Revista de lepidopterolog\u00eda","issn_l":"0300-5267","issn":["0300-5267","2340-4078"],"is_oa":true,"is_in_doaj":true,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Algorithms, Vol 18, Iss 6, p 334 (2025)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.3390/a18060334","is_oa":true,"landing_page_url":"https://doi.org/10.3390/a18060334","pdf_url":"https://www.mdpi.com/1999-4893/18/6/334/pdf?version=1748931663","source":{"id":"https://openalex.org/S190629608","display_name":"Algorithms","issn_l":"1999-4893","issn":["1999-4893"],"is_oa":true,"is_in_doaj":true,"is_core":true,"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":"Algorithms","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4411002368.pdf","grobid_xml":"https://content.openalex.org/works/W4411002368.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W1500046813","https://openalex.org/W1966713076","https://openalex.org/W1968365214","https://openalex.org/W2032251455","https://openalex.org/W2039888026","https://openalex.org/W2053312457","https://openalex.org/W2133500334","https://openalex.org/W2890802844","https://openalex.org/W2899283552","https://openalex.org/W3175846571","https://openalex.org/W3214177313","https://openalex.org/W3217139884","https://openalex.org/W4250867923","https://openalex.org/W4309548460","https://openalex.org/W4362674419","https://openalex.org/W4381687077","https://openalex.org/W4386936019","https://openalex.org/W4390549177","https://openalex.org/W4401971520","https://openalex.org/W4402774937","https://openalex.org/W4404235276","https://openalex.org/W6851805877","https://openalex.org/W6860172205"],"related_works":["https://openalex.org/W2391251536","https://openalex.org/W2362198218","https://openalex.org/W2019521278","https://openalex.org/W1984922432","https://openalex.org/W2375008505","https://openalex.org/W1982750869","https://openalex.org/W2085756966","https://openalex.org/W1734603769","https://openalex.org/W2922935260","https://openalex.org/W1592131421"],"abstract_inverted_index":{"In":[0,66],"this":[1],"work,":[2],"the":[3,28,34,52,57,67,70,76,80,86,94,108,122,143,146,156],"construction":[4],"of":[5,36,96,110,142,158],"adaptive":[6],"one-dimensional":[7],"computational":[8],"grids":[9],"is":[10,25,73,116,148],"considered":[11],"using":[12],"a":[13,40,135],"numerical":[14,44,123,174],"method":[15,45,124],"and":[16,107,164],"physics-informed":[17],"neural":[18,81],"networks":[19],"(PINNs).":[20],"The":[21,43,118,150],"grid":[22,87,114,137,159],"adaptation":[23,115,160],"process":[24],"described":[26],"by":[27],"diffusion":[29,111],"equation,":[30],"which":[31],"allows":[32],"for":[33,154,165,169],"redistribution":[35],"nodes":[37],"based":[38,88],"on":[39,89,113],"control":[41,104],"function.":[42],"employs":[46],"an":[47],"iterative":[48],"scheme":[49],"that":[50,121],"adjusts":[51],"node":[53,130],"positions":[54],"according":[55],"to":[56,84,126],"solution":[58],"gradients,":[59],"ensuring":[60],"local":[61],"refinement":[62],"in":[63,129,161,173],"key":[64],"regions.":[65],"PINN":[68],"approach,":[69],"governing":[71],"equation":[72],"incorporated":[74],"into":[75],"loss":[77],"function,":[78],"enabling":[79],"network":[82],"model":[83],"generate":[85],"physical":[90],"constraints.":[91],"To":[92],"evaluate":[93],"performance":[95],"both":[97,162],"methods,":[98],"tests":[99],"are":[100],"conducted":[101],"with":[102],"different":[103],"function":[105],"parameters,":[106],"influence":[109],"coefficients":[112],"analyzed.":[117],"results":[119],"show":[120],"leads":[125],"sharper":[127],"variations":[128],"spacing,":[131],"while":[132],"PINNs":[133],"produce":[134],"smoother":[136],"distribution.":[138],"A":[139],"comparative":[140],"analysis":[141],"deviations":[144],"between":[145],"methods":[147],"performed.":[149],"obtained":[151],"data":[152],"allow":[153],"assessing":[155],"characteristics":[157],"approaches":[163],"identifying":[166],"possible":[167],"directions":[168],"their":[170],"further":[171],"application":[172],"modeling":[175],"tasks.":[176]},"counts_by_year":[],"updated_date":"2026-03-10T16:38:18.471706","created_date":"2025-10-10T00:00:00"}
