{"id":"https://openalex.org/W4414971137","doi":"https://doi.org/10.48550/arxiv.2510.04170","title":"Robust and efficient solvers for nonlinear partial differential equations based on random feature method","display_name":"Robust and efficient solvers for nonlinear partial differential equations based on random feature method","publication_year":2025,"publication_date":"2025-10-05","ids":{"openalex":"https://openalex.org/W4414971137","doi":"https://doi.org/10.48550/arxiv.2510.04170"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2510.04170","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2510.04170","pdf_url":"https://arxiv.org/pdf/2510.04170","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"type":"preprint","indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2510.04170","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":null,"display_name":"Tan, Longze","orcid":null},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Tan, Longze","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"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/T14257","display_name":"Advanced Measurement and Detection Methods","score":0.5616000294685364,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T14257","display_name":"Advanced Measurement and Detection Methods","score":0.5616000294685364,"subfield":{"id":"https://openalex.org/subfields/2208","display_name":"Electrical and Electronic 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/T13717","display_name":"Advanced Algorithms and Applications","score":0.5400000214576721,"subfield":{"id":"https://openalex.org/subfields/2207","display_name":"Control and Systems 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/T13832","display_name":"Advanced Decision-Making Techniques","score":0.5148000121116638,"subfield":{"id":"https://openalex.org/subfields/1710","display_name":"Information Systems"},"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/preconditioner","display_name":"Preconditioner","score":0.7860000133514404},{"id":"https://openalex.org/keywords/jacobian-matrix-and-determinant","display_name":"Jacobian matrix and determinant","score":0.6811000108718872},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5831999778747559},{"id":"https://openalex.org/keywords/discretization","display_name":"Discretization","score":0.5684999823570251},{"id":"https://openalex.org/keywords/residual","display_name":"Residual","score":0.5358999967575073},{"id":"https://openalex.org/keywords/quasi-newton-method","display_name":"Quasi-Newton method","score":0.4235000014305115},{"id":"https://openalex.org/keywords/newtons-method","display_name":"Newton's method","score":0.3806999921798706},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.3734999895095825},{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.3564000129699707}],"concepts":[{"id":"https://openalex.org/C167431342","wikidata":"https://www.wikidata.org/wiki/Q1754327","display_name":"Preconditioner","level":3,"score":0.7860000133514404},{"id":"https://openalex.org/C200331156","wikidata":"https://www.wikidata.org/wiki/Q506041","display_name":"Jacobian matrix and determinant","level":2,"score":0.6811000108718872},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5831999778747559},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.58160001039505},{"id":"https://openalex.org/C73000952","wikidata":"https://www.wikidata.org/wiki/Q17007827","display_name":"Discretization","level":2,"score":0.5684999823570251},{"id":"https://openalex.org/C155512373","wikidata":"https://www.wikidata.org/wiki/Q287450","display_name":"Residual","level":2,"score":0.5358999967575073},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.45660001039505005},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.43689998984336853},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.42649999260902405},{"id":"https://openalex.org/C114954040","wikidata":"https://www.wikidata.org/wiki/Q1476018","display_name":"Quasi-Newton method","level":4,"score":0.4235000014305115},{"id":"https://openalex.org/C85189116","wikidata":"https://www.wikidata.org/wiki/Q374195","display_name":"Newton's method","level":3,"score":0.3806999921798706},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.3734999895095825},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3619999885559082},{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.3564000129699707},{"id":"https://openalex.org/C203616005","wikidata":"https://www.wikidata.org/wiki/Q620495","display_name":"Hessian matrix","level":2,"score":0.3555999994277954},{"id":"https://openalex.org/C2776401178","wikidata":"https://www.wikidata.org/wiki/Q12050496","display_name":"Feature (linguistics)","level":2,"score":0.3418999910354614},{"id":"https://openalex.org/C36503486","wikidata":"https://www.wikidata.org/wiki/Q11235244","display_name":"Domain (mathematical analysis)","level":2,"score":0.33500000834465027},{"id":"https://openalex.org/C85522705","wikidata":"https://www.wikidata.org/wiki/Q3278015","display_name":"Line search","level":3,"score":0.3246999979019165},{"id":"https://openalex.org/C111335779","wikidata":"https://www.wikidata.org/wiki/Q3454686","display_name":"Reduction (mathematics)","level":2,"score":0.31949999928474426},{"id":"https://openalex.org/C80023036","wikidata":"https://www.wikidata.org/wiki/Q5147531","display_name":"Collocation (remote sensing)","level":2,"score":0.31690001487731934},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.31049999594688416},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.3100999891757965},{"id":"https://openalex.org/C6802819","wikidata":"https://www.wikidata.org/wiki/Q1072174","display_name":"Linear system","level":2,"score":0.30880001187324524},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.3027999997138977},{"id":"https://openalex.org/C48753275","wikidata":"https://www.wikidata.org/wiki/Q11216","display_name":"Numerical analysis","level":2,"score":0.29010000824928284},{"id":"https://openalex.org/C63479239","wikidata":"https://www.wikidata.org/wiki/Q7353546","display_name":"Robustness (evolution)","level":3,"score":0.28130000829696655},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.27559998631477356},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.27309998869895935},{"id":"https://openalex.org/C2776003309","wikidata":"https://www.wikidata.org/wiki/Q1988072","display_name":"Adaptive algorithm","level":2,"score":0.2687000036239624},{"id":"https://openalex.org/C32230216","wikidata":"https://www.wikidata.org/wiki/Q7882499","display_name":"Uncertainty quantification","level":2,"score":0.2639000117778778}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2510.04170","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2510.04170","pdf_url":"https://arxiv.org/pdf/2510.04170","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2510.04170","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2510.04170","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":"article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2510.04170","is_oa":true,"landing_page_url":"http://arxiv.org/abs/2510.04170","pdf_url":"https://arxiv.org/pdf/2510.04170","source":{"id":"https://openalex.org/S4393918464","display_name":"ArXiv.org","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":[],"abstract_inverted_index":{"The":[0,64],"random":[1,40],"feature":[2],"method":[3,70,132],"(RFM),":[4],"a":[5,13,106,147],"mesh-free":[6],"machine":[7],"learning-based":[8],"framework,":[9,122],"has":[10],"emerged":[11],"as":[12,229],"promising":[14],"alternative":[15],"for":[16,23,233],"solving":[17],"PDEs":[18,191],"on":[19,184],"complex":[20,193],"domains.":[21],"However,":[22],"large":[24,48],"three-dimensional":[25,186],"nonlinear":[26,51,235],"problems,":[27],"attaining":[28],"high":[29],"accuracy":[30,217],"typically":[31],"requires":[32],"domain":[33],"partitioning":[34],"with":[35,71,155,192,204,227],"many":[36],"collocation":[37],"points":[38],"and":[39,49,80,105,116,178,188,208,219],"features":[41],"per":[42],"subdomain,":[43],"which":[44,76],"leads":[45],"to":[46,85,112],"extremely":[47],"ill-conditioned":[50],"least-squares":[52],"systems.":[53],"To":[54],"overcome":[55],"these":[56],"challenges,":[57],"we":[58,123],"propose":[59],"two":[60],"randomized":[61,77],"Newton-type":[62],"solvers.":[63,202],"first":[65],"is":[66,99,110,140],"an":[67,87,126,156,230],"inexact":[68,129],"Newton":[69,97,131],"right":[72],"preconditioning":[73],"(IPN),":[74],"in":[75,168],"Jacobian":[78,139],"compression":[79],"QR":[81],"factorization":[82],"are":[83],"used":[84],"construct":[86],"efficient":[88,231],"preconditioner":[89,164],"that":[90],"substantially":[91],"reduces":[92],"the":[93,137,162,196,200,212,224],"condition":[94],"number.":[95],"Each":[96],"step":[98],"then":[100],"approximately":[101],"solved":[102],"by":[103],"LSQR,":[104],"derivative-free":[107],"line":[108],"search":[109],"incorporated":[111],"ensure":[113],"residual":[114],"reduction":[115],"stable":[117],"convergence.":[118],"Building":[119],"upon":[120],"this":[121,135],"further":[124],"develop":[125],"adaptive":[127,157],"multi-step":[128],"preconditioned":[130,138],"(AMIPN).":[133],"In":[134],"approach,":[136],"reused":[141],"across":[142],"multiple":[143],"inner":[144,152],"iterations,":[145],"while":[146],"prescribed":[148],"maximum":[149],"number":[150],"of":[151,199],"iterations":[153],"together":[154],"early-stopping":[158],"criterion":[159],"determines":[160],"whether":[161],"current":[163],"can":[165],"be":[166],"retained":[167],"subsequent":[169],"outer":[170],"iterations.":[171],"These":[172],"mechanisms":[173],"effectively":[174],"avoid":[175],"redundant":[176],"computations":[177],"enhance":[179],"robustness.":[180],"Extensive":[181],"numerical":[182],"experiments":[183],"both":[185],"steady-state":[187],"two-dimensional":[189],"time-dependent":[190],"geometries":[194],"confirm":[195],"remarkable":[197],"effectiveness":[198],"proposed":[201],"Compared":[203],"classical":[205],"discretization":[206],"techniques":[207],"recent":[209],"machine-learning-based":[210],"approaches,":[211],"methods":[213],"consistently":[214],"deliver":[215],"substantial":[216],"improvements":[218],"robust":[220],"convergence,":[221],"thereby":[222],"establishing":[223],"RFM":[225],"combined":[226],"IPN/AMIPN":[228],"framework":[232],"large-scale":[234],"PDEs.":[236],".":[237]},"counts_by_year":[],"updated_date":"2026-04-16T08:26:57.006410","created_date":"2025-10-09T00:00:00"}
