{"id":"https://openalex.org/W3174797791","doi":"https://doi.org/10.1109/access.2021.3091590","title":"Optimizing the Coefficients of Numerical Differentiation Formulae Using Neural Networks","display_name":"Optimizing the Coefficients of Numerical Differentiation Formulae Using Neural Networks","publication_year":2021,"publication_date":"2021-01-01","ids":{"openalex":"https://openalex.org/W3174797791","doi":"https://doi.org/10.1109/access.2021.3091590","mag":"3174797791"},"language":"en","primary_location":{"id":"doi:10.1109/access.2021.3091590","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2021.3091590","pdf_url":"https://ieeexplore.ieee.org/ielx7/6287639/9312710/09462164.pdf","source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Access","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://ieeexplore.ieee.org/ielx7/6287639/9312710/09462164.pdf","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101840363","display_name":"Xinyu Yang","orcid":"https://orcid.org/0000-0001-8143-6911"},"institutions":[{"id":"https://openalex.org/I173899330","display_name":"Henan University","ror":"https://ror.org/003xyzq10","country_code":"CN","type":"education","lineage":["https://openalex.org/I173899330"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Xinyu Yang","raw_affiliation_strings":["School of Physics and Electronics, Henan University, Kaifeng, China"],"raw_orcid":"https://orcid.org/0000-0001-8143-6911","affiliations":[{"raw_affiliation_string":"School of Physics and Electronics, Henan University, Kaifeng, China","institution_ids":["https://openalex.org/I173899330"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100637601","display_name":"Lei Zhang","orcid":"https://orcid.org/0000-0002-8481-5914"},"institutions":[{"id":"https://openalex.org/I173899330","display_name":"Henan University","ror":"https://ror.org/003xyzq10","country_code":"CN","type":"education","lineage":["https://openalex.org/I173899330"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Lei Zhang","raw_affiliation_strings":["School of Physics and Electronics, Henan University, Kaifeng, China"],"raw_orcid":"https://orcid.org/0000-0002-8481-5914","affiliations":[{"raw_affiliation_string":"School of Physics and Electronics, Henan University, Kaifeng, China","institution_ids":["https://openalex.org/I173899330"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100334736","display_name":"Yuan Yuan","orcid":"https://orcid.org/0000-0001-9974-0516"},"institutions":[{"id":"https://openalex.org/I173899330","display_name":"Henan University","ror":"https://ror.org/003xyzq10","country_code":"CN","type":"education","lineage":["https://openalex.org/I173899330"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Yuan Yuan","raw_affiliation_strings":["School of Physics and Electronics, Henan University, Kaifeng, China"],"raw_orcid":"https://orcid.org/0000-0001-9974-0516","affiliations":[{"raw_affiliation_string":"School of Physics and Electronics, Henan University, Kaifeng, China","institution_ids":["https://openalex.org/I173899330"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5040798310","display_name":"Zhiqi Zhao","orcid":"https://orcid.org/0000-0001-6319-5114"},"institutions":[{"id":"https://openalex.org/I173899330","display_name":"Henan University","ror":"https://ror.org/003xyzq10","country_code":"CN","type":"education","lineage":["https://openalex.org/I173899330"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Zhiqi Zhao","raw_affiliation_strings":["School of Physics and Electronics, Henan University, Kaifeng, China"],"raw_orcid":"https://orcid.org/0000-0001-6319-5114","affiliations":[{"raw_affiliation_string":"School of Physics and Electronics, Henan University, Kaifeng, China","institution_ids":["https://openalex.org/I173899330"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I173899330"],"apc_list":{"value":1850,"currency":"USD","value_usd":1850},"apc_paid":{"value":1850,"currency":"USD","value_usd":1850},"fwci":0.3072,"has_fulltext":true,"cited_by_count":5,"citation_normalized_percentile":{"value":0.52309099,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":97},"biblio":{"volume":"9","issue":null,"first_page":"92113","last_page":"92124"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9968000054359436,"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"}},"topics":[{"id":"https://openalex.org/T11416","display_name":"Numerical methods for differential equations","score":0.9968000054359436,"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"}},{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9930999875068665,"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/T10792","display_name":"Matrix Theory and Algorithms","score":0.991599977016449,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/computer-science","display_name":"Computer science","score":0.6332197785377502},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5927222967147827},{"id":"https://openalex.org/keywords/numerical-differentiation","display_name":"Numerical differentiation","score":0.4160485565662384},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.25469034910202026},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.2269650101661682},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.12388962507247925}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.6332197785377502},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5927222967147827},{"id":"https://openalex.org/C154181440","wikidata":"https://www.wikidata.org/wiki/Q1683292","display_name":"Numerical differentiation","level":2,"score":0.4160485565662384},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.25469034910202026},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.2269650101661682},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.12388962507247925}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/access.2021.3091590","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2021.3091590","pdf_url":"https://ieeexplore.ieee.org/ielx7/6287639/9312710/09462164.pdf","source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Access","raw_type":"journal-article"},{"id":"pmh:oai:doaj.org/article:f30a556367ec47d890ecc4b72d792111","is_oa":true,"landing_page_url":"https://doaj.org/article/f30a556367ec47d890ecc4b72d792111","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-sa","license_id":"https://openalex.org/licenses/cc-by-sa","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"IEEE Access, Vol 9, Pp 92113-92124 (2021)","raw_type":"article"}],"best_oa_location":{"id":"doi:10.1109/access.2021.3091590","is_oa":true,"landing_page_url":"https://doi.org/10.1109/access.2021.3091590","pdf_url":"https://ieeexplore.ieee.org/ielx7/6287639/9312710/09462164.pdf","source":{"id":"https://openalex.org/S2485537415","display_name":"IEEE Access","issn_l":"2169-3536","issn":["2169-3536"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Access","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3774283542","display_name":null,"funder_award_id":"61773008","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W3174797791.pdf","grobid_xml":"https://content.openalex.org/works/W3174797791.grobid-xml"},"referenced_works_count":46,"referenced_works":["https://openalex.org/W102596075","https://openalex.org/W280536263","https://openalex.org/W1498436455","https://openalex.org/W1558016992","https://openalex.org/W1974032685","https://openalex.org/W1977106683","https://openalex.org/W1985262583","https://openalex.org/W1995504856","https://openalex.org/W2005653523","https://openalex.org/W2020333131","https://openalex.org/W2020346173","https://openalex.org/W2020747585","https://openalex.org/W2049688341","https://openalex.org/W2050630313","https://openalex.org/W2071305979","https://openalex.org/W2078767954","https://openalex.org/W2081953057","https://openalex.org/W2087070363","https://openalex.org/W2090322886","https://openalex.org/W2092716200","https://openalex.org/W2103086012","https://openalex.org/W2117453669","https://openalex.org/W2138472326","https://openalex.org/W2146194206","https://openalex.org/W2154876043","https://openalex.org/W2155809947","https://openalex.org/W2166827401","https://openalex.org/W2183174234","https://openalex.org/W2481944677","https://openalex.org/W2740885083","https://openalex.org/W2757947589","https://openalex.org/W2764226165","https://openalex.org/W2770146298","https://openalex.org/W2791527216","https://openalex.org/W2800106114","https://openalex.org/W2877622484","https://openalex.org/W2897025162","https://openalex.org/W2905562343","https://openalex.org/W2908245690","https://openalex.org/W2986587829","https://openalex.org/W2990048388","https://openalex.org/W3016981486","https://openalex.org/W3093347455","https://openalex.org/W4251074877","https://openalex.org/W4255754126","https://openalex.org/W6681642280"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W2088429582","https://openalex.org/W2390279801","https://openalex.org/W4391913857","https://openalex.org/W2358668433","https://openalex.org/W4396701345","https://openalex.org/W2376932109","https://openalex.org/W2001405890"],"abstract_inverted_index":{"The":[0,57,102,140],"use":[1],"of":[2,54,59,61,73,93,119,134,159],"a":[3,38],"numerical":[4,136,146],"differentiation":[5],"formula":[6],"(NDF)":[7],"is":[8,42,100,106,149,166],"an":[9],"excellent":[10],"method":[11,21,32],"for":[12,33,51,169],"solving":[13,170],"stiff":[14,27,55,81,121,171],"ordinary":[15,172],"differential":[16,173],"equations.":[17,56,174],"However,":[18],"the":[19,62,71,79,85,88,91,94,110,116,125,128,135,145,153,157,160,163],"NDF":[20,35,86,98,126,154],"cannot":[22],"fully":[23],"adapt":[24],"to":[25,64,77,108],"all":[26],"systems.":[28,122],"An":[29],"optimal":[30,117],"general":[31],"optimizing":[34,152],"coefficients":[36,66,99,155],"using":[37],"back-propagation":[39,103],"neural":[40,104],"network":[41,105],"proposed":[43],"in":[44],"this":[45],"work":[46],"that":[47,144],"can":[48],"be":[49],"used":[50,107],"different":[52,80,120],"systems":[53,82],"ranges":[58],"stability":[60],"first-":[63],"fourth-order":[65],"are":[67,138],"obtained":[68],"by":[69,83],"analyzing":[70],"definition":[72],"stability.":[74],"In":[75],"order":[76],"solve":[78],"changing":[84],"coefficients,":[87,127],"relationship":[89,111],"between":[90,112],"eigenvalues":[92],"Jacobian":[95],"matrix":[96],"and":[97,114,130],"analyzed.":[101],"describe":[109],"them":[113],"predict":[115],"parameters":[118],"Compared":[123],"with":[124],"absolute":[129],"mean":[131],"square":[132],"errors":[133],"solutions":[137],"smaller.":[139],"simulation":[141,161],"results":[142],"show":[143],"solution\u2019s":[147],"accuracy":[148,158],"higher.":[150],"Because":[151],"improves":[156],"results,":[162],"new":[164],"approach":[165],"more":[167],"suitable":[168]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}