{"id":"https://openalex.org/W4401521126","doi":"https://doi.org/10.1137/23m1568107","title":"Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: Discontinuity Interface","display_name":"Least-Squares Neural Network (LSNN) Method for Linear Advection-Reaction Equation: Discontinuity Interface","publication_year":2024,"publication_date":"2024-08-12","ids":{"openalex":"https://openalex.org/W4401521126","doi":"https://doi.org/10.1137/23m1568107"},"language":"en","primary_location":{"id":"doi:10.1137/23m1568107","is_oa":false,"landing_page_url":"https://doi.org/10.1137/23m1568107","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"},"type":"article","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/A5023481731","display_name":"Zhiqiang Cai","orcid":"https://orcid.org/0000-0002-3775-7248"},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Zhiqiang Cai","raw_affiliation_strings":["Department of Mathematics, Purdue University, West Lafayette, IN 47907-2067 USA","Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067"],"raw_orcid":"https://orcid.org/0000-0002-3775-7248","affiliations":[{"raw_affiliation_string":"Department of Mathematics, Purdue University, West Lafayette, IN 47907-2067 USA","institution_ids":["https://openalex.org/I219193219"]},{"raw_affiliation_string":"Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067","institution_ids":["https://openalex.org/I219193219"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079481968","display_name":"Junpyo Choi","orcid":null},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Junpyo Choi","raw_affiliation_strings":["Department of Mathematics, Purdue University, West Lafayette, IN 47907-2067 USA","Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Purdue University, West Lafayette, IN 47907-2067 USA","institution_ids":["https://openalex.org/I219193219"]},{"raw_affiliation_string":"Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN 47907-2067","institution_ids":["https://openalex.org/I219193219"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100343854","display_name":"Min Liu","orcid":"https://orcid.org/0000-0001-6406-4896"},"institutions":[{"id":"https://openalex.org/I219193219","display_name":"Purdue University West Lafayette","ror":"https://ror.org/02dqehb95","country_code":"US","type":"education","lineage":["https://openalex.org/I219193219"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Min Liu","raw_affiliation_strings":["School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088 USA","School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088 USA","institution_ids":["https://openalex.org/I219193219"]},{"raw_affiliation_string":"School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907-2088","institution_ids":["https://openalex.org/I219193219"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5023481731"],"corresponding_institution_ids":["https://openalex.org/I219193219"],"apc_list":null,"apc_paid":null,"fwci":0.8087,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.70606898,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":95,"max":98},"biblio":{"volume":"46","issue":"4","first_page":"C448","last_page":"C478"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11206","display_name":"Model Reduction and Neural Networks","score":0.9994000196456909,"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":0.9994000196456909,"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/T12560","display_name":"Nuclear Engineering Thermal-Hydraulics","score":0.9886999726295471,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"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/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9761999845504761,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/discontinuity","display_name":"Discontinuity (linguistics)","score":0.6668441295623779},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6345037221908569},{"id":"https://openalex.org/keywords/advection","display_name":"Advection","score":0.5652753114700317},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.5469480156898499},{"id":"https://openalex.org/keywords/linear-equation","display_name":"Linear equation","score":0.48483335971832275},{"id":"https://openalex.org/keywords/interface","display_name":"Interface (matter)","score":0.434885710477829},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4298152029514313},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3833353519439697},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3363436162471771},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.19336730241775513},{"id":"https://openalex.org/keywords/mechanics","display_name":"Mechanics","score":0.15369558334350586},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.1386616826057434},{"id":"https://openalex.org/keywords/thermodynamics","display_name":"Thermodynamics","score":0.13258254528045654},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12042221426963806}],"concepts":[{"id":"https://openalex.org/C2777042112","wikidata":"https://www.wikidata.org/wiki/Q5281658","display_name":"Discontinuity (linguistics)","level":2,"score":0.6668441295623779},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6345037221908569},{"id":"https://openalex.org/C5072599","wikidata":"https://www.wikidata.org/wiki/Q379788","display_name":"Advection","level":2,"score":0.5652753114700317},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.5469480156898499},{"id":"https://openalex.org/C77346573","wikidata":"https://www.wikidata.org/wiki/Q484637","display_name":"Linear equation","level":2,"score":0.48483335971832275},{"id":"https://openalex.org/C113843644","wikidata":"https://www.wikidata.org/wiki/Q901882","display_name":"Interface (matter)","level":4,"score":0.434885710477829},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4298152029514313},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3833353519439697},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3363436162471771},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.19336730241775513},{"id":"https://openalex.org/C57879066","wikidata":"https://www.wikidata.org/wiki/Q41217","display_name":"Mechanics","level":1,"score":0.15369558334350586},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.1386616826057434},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.13258254528045654},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12042221426963806},{"id":"https://openalex.org/C157915830","wikidata":"https://www.wikidata.org/wiki/Q2928001","display_name":"Bubble","level":2,"score":0.0},{"id":"https://openalex.org/C129307140","wikidata":"https://www.wikidata.org/wiki/Q6795880","display_name":"Maximum bubble pressure method","level":3,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1137/23m1568107","is_oa":false,"landing_page_url":"https://doi.org/10.1137/23m1568107","pdf_url":null,"source":{"id":"https://openalex.org/S165512578","display_name":"SIAM Journal on Scientific Computing","issn_l":"1064-8275","issn":["1064-8275","1095-7197"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Scientific Computing","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Climate action","id":"https://metadata.un.org/sdg/13","score":0.4399999976158142}],"awards":[{"id":"https://openalex.org/G8080588884","display_name":null,"funder_award_id":"DMS-2110571","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"}],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":33,"referenced_works":["https://openalex.org/W1970454553","https://openalex.org/W2006910694","https://openalex.org/W2031978570","https://openalex.org/W2040628885","https://openalex.org/W2065787176","https://openalex.org/W2069930283","https://openalex.org/W2139868442","https://openalex.org/W2158581396","https://openalex.org/W2158985775","https://openalex.org/W2559827045","https://openalex.org/W2749028154","https://openalex.org/W2760972773","https://openalex.org/W2770250658","https://openalex.org/W2891039272","https://openalex.org/W2899283552","https://openalex.org/W2949883708","https://openalex.org/W2963146412","https://openalex.org/W2985622559","https://openalex.org/W3016664815","https://openalex.org/W3045456967","https://openalex.org/W3047245470","https://openalex.org/W3103541433","https://openalex.org/W3130671418","https://openalex.org/W3164919902","https://openalex.org/W3171209985","https://openalex.org/W3179979867","https://openalex.org/W3183891129","https://openalex.org/W3194828218","https://openalex.org/W3196900140","https://openalex.org/W4297797509","https://openalex.org/W4367840231","https://openalex.org/W4385143762","https://openalex.org/W4385431227"],"related_works":["https://openalex.org/W3036101264","https://openalex.org/W2163151373","https://openalex.org/W1986676657","https://openalex.org/W4238257821","https://openalex.org/W2165001118","https://openalex.org/W2022713607","https://openalex.org/W1702821","https://openalex.org/W2794551621","https://openalex.org/W1484056900","https://openalex.org/W4246396821"],"abstract_inverted_index":{"We":[0],"studied":[1],"the":[2,65,74,90,135,149,169,173,177,195,207],"least-squares":[3,45],"ReLU":[4,25,55,101,153],"neural":[5,26,56],"network":[6,27,57],"(LSNN)":[7,28],"method":[8,29,40,92,151,197],"for":[9,30,93,183],"solving":[10],"linear":[11,31],"advectionreaction":[12],"equation":[13],"with":[14,95,104,130,156,189,198],"discontinuous":[15,140],"solution":[16,136,174,178],"in":[17],"[Cai,":[18],"Zhiqiang,":[19],"Jingshuang":[20],"Chen,":[21],"and":[22,47,61,80,139,203],"Min":[23],"Liu.\"Least-squares":[24],"advection-reaction":[32],"equation.\"Journal":[33],"of":[34,52,64,76,148,172],"Computational":[35],"Physics":[36],"443":[37],"(2021),":[38],"110514].The":[39],"is":[41,73,165,180,201],"based":[42],"on":[43,119],"a":[44,49,77,120,125],"formulation":[46],"uses":[48],"new":[50],"class":[51],"approximating":[53],"functions:":[54],"(NN)":[58],"functions.A":[59],"critical":[60],"additional":[62],"component":[63],"LSNN":[66,91,150,196],"method,":[67],"differing":[68],"from":[69],"other":[70],"NN-based":[71],"methods,":[72],"introduction":[75],"properly":[78],"designed":[79],"physics":[81],"preserved":[82],"discrete":[83],"differential":[84],"operator.In":[85],"this":[86],"paper,":[87],"we":[88,98,142],"study":[89],"problems":[94,188],"discontinuity":[96,121,191,212],"interfaces.First,":[97],"show":[99,193],"that":[100,145,176,194],"NN":[102,154],"functions":[103,155],"depth":[105,157],"\u2308log":[106,158],"2":[107,159],"(d":[108,160],"+":[109,111,161,163],"1)\u2309":[110,162],"1":[112,164],"can":[113],"approximate":[114],"any":[115,131],"d-dimensional":[116],"step":[117],"function":[118],"interface":[122],"generated":[123],"by":[124,168],"vector":[126],"field":[127],"as":[128],"streamlines":[129],"prescribed":[132],"accuracy.By":[133],"decomposing":[134],"into":[137],"continuous":[138,170],"parts,":[141],"prove":[143],"theoretically":[144],"discretization":[146],"error":[147],"using":[152],"mainly":[166],"determined":[167],"part":[171],"provided":[175],"jump":[179],"constant.Numerical":[181],"results":[182],"both":[184],"two-and":[185],"three-dimensional":[186],"test":[187],"various":[190],"interfaces":[192],"enough":[199],"layers":[200],"accurate":[202],"does":[204],"not":[205],"exhibit":[206],"common":[208],"Gibbs":[209],"phenomena":[210],"along":[211],"interfaces.":[213]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":2}],"updated_date":"2025-12-23T23:11:35.936235","created_date":"2024-08-13T00:00:00"}
