Neural network approach for solving nonlocal boundary value problems

This paper proposes a radial basis function (RBF) network-based method for solving a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the differential equation needs a completely different approach from the up-to-now-known methods for solving...

Celý popis

Uložené v:

Podrobná bibliografia
Vydané v:	Neural computing & applications Ročník 32; číslo 17; s. 14153 - 14171
Hlavní autori:	Palade, V., Petrov, M. S., Todorov, T. D.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	London Springer London 01.09.2020 Springer Nature B.V
Predmet:	Artificial Intelligence Boundary conditions Boundary value problems Computational Biology/Bioinformatics Computational Science and Engineering Computer Science Data Mining and Knowledge Discovery Differential equations Dirichlet problem Divergence Finite element method Image Processing and Computer Vision Learning Mathematical analysis Methods Neural networks Numerical integration Original Article Probability and Statistics in Computer Science Radial basis function Variational methods RBF neural network Nonlocal nonlinear problem Two-point step size gradient method Variable learning rate
ISSN:	0941-0643, 1433-3058
On-line prístup:	Získať plný text
Tagy:	Pridať tag Žiadne tagy, Buďte prvý, kto otaguje tento záznam!

Abstract	This paper proposes a radial basis function (RBF) network-based method for solving a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the differential equation needs a completely different approach from the up-to-now-known methods for solving boundary value problems by using neural networks. A numerical integration procedure is developed for computing the local L 2 -inner product. It is known that the non-variational methods are not effective in solving nonlocal problems. In this paper, the weak formulation of the nonlocal problem is reduced to the minimization of a nonlinear functional. Unlike many previous works, we use an integral objective functional for implementing the learning procedure. Well-distributed nodes are used as the centers of the RBF neural network. The weights of the RBF network are determined by a two-point step size gradient method. The neural network method proposed in this paper is an alternative to the finite-element method (FEM) for solving nonlocal boundary value problems in non-Lipschitz domains. A new variable learning rate strategy has been developed and implemented in order to avoid the divergence of the training process. A comparison between the proposed neural network approach and the FEM is illustrated by challenging examples, and the performance of both methods is thoroughly analyzed.
AbstractList	This paper proposes a radial basis function (RBF) network-based method for solving a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the differential equation needs a completely different approach from the up-to-now-known methods for solving boundary value problems by using neural networks. A numerical integration procedure is developed for computing the local L 2 -inner product. It is known that the non-variational methods are not effective in solving nonlocal problems. In this paper, the weak formulation of the nonlocal problem is reduced to the minimization of a nonlinear functional. Unlike many previous works, we use an integral objective functional for implementing the learning procedure. Well-distributed nodes are used as the centers of the RBF neural network. The weights of the RBF network are determined by a two-point step size gradient method. The neural network method proposed in this paper is an alternative to the finite-element method (FEM) for solving nonlocal boundary value problems in non-Lipschitz domains. A new variable learning rate strategy has been developed and implemented in order to avoid the divergence of the training process. A comparison between the proposed neural network approach and the FEM is illustrated by challenging examples, and the performance of both methods is thoroughly analyzed. This paper proposes a radial basis function (RBF) network-based method for solving a nonlinear second-order elliptic equation with Dirichlet boundary conditions. The nonlocal term involved in the differential equation needs a completely different approach from the up-to-now-known methods for solving boundary value problems by using neural networks. A numerical integration procedure is developed for computing the local L2-inner product. It is known that the non-variational methods are not effective in solving nonlocal problems. In this paper, the weak formulation of the nonlocal problem is reduced to the minimization of a nonlinear functional. Unlike many previous works, we use an integral objective functional for implementing the learning procedure. Well-distributed nodes are used as the centers of the RBF neural network. The weights of the RBF network are determined by a two-point step size gradient method. The neural network method proposed in this paper is an alternative to the finite-element method (FEM) for solving nonlocal boundary value problems in non-Lipschitz domains. A new variable learning rate strategy has been developed and implemented in order to avoid the divergence of the training process. A comparison between the proposed neural network approach and the FEM is illustrated by challenging examples, and the performance of both methods is thoroughly analyzed.
Author	Petrov, M. S. Todorov, T. D. Palade, V.
Author_xml	– sequence: 1 givenname: V. surname: Palade fullname: Palade, V. organization: Faculty of Engineering and Computing, Coventry University – sequence: 2 givenname: M. S. surname: Petrov fullname: Petrov, M. S. organization: Department of Technical Mechanics, Technical University of Gabrovo – sequence: 3 givenname: T. D. surname: Todorov fullname: Todorov, T. D. email: tdtodorov@tugab.bg organization: Department of Mathematics and Informatics, Technical University of Gabrovo
BookMark	eNp9kEtPwzAQhC1UJNrCH-AUiXNg_Yx9ROUpIbjA2XITu6SkdrGTIv49hiAhcehl97Dz7YxmhiY-eIvQKYZzDFBdJABOcAkESmAS53mApphRWlLgcoKmoFg-C0aP0CylNQAwIfkUXT3aIZqu8Lb_CPGtMNttDKZ-LVyIRQrdrvWrIpt1oc6qZRh8Y-JnsTPdYIssXXZ2k47RoTNdsie_e45ebq6fF3flw9Pt_eLyoawpVn1pDCZKcCsJqIbaumKGy0YtmcFKVFJycLIWDVOc5dSACTPWOSWN5MJW4OgcnY1_s_H7YFOv12GIPltqwigTmOCKZRUZVXUMKUXr9Da2m5xaY9DfbemxLZ3b0j9taciQ_AfVbW_6Nvg-mrbbj9IRTdnHr2z8S7WH-gJGtH_J
CitedBy_id	crossref_primary_10_1016_j_camwa_2023_04_026 crossref_primary_10_1016_j_heliyon_2023_e23439 crossref_primary_10_1109_ACCESS_2020_3048104 crossref_primary_10_3390_math11183935 crossref_primary_10_1016_j_rineng_2025_105114 crossref_primary_10_1038_s41598_021_94331_0 crossref_primary_10_1109_ACCESS_2025_3598376
Cites_doi	10.1137/110846397 10.1016/j.apnum.2005.11.007 10.1007/BF03177517 10.1016/j.cnsns.2007.04.024 10.1016/j.cam.2017.06.022 10.1016/j.advengsoft.2009.06.003 10.1002/cnm.522 10.1016/j.camwa.2018.06.019 10.14232/ejqtde.2012.1.58 10.21914/anziamj.v42i0.611 10.2307/2002483 10.1016/j.neucom.2008.12.004 10.1016/j.cam.2015.02.048 10.1016/j.camwa.2013.07.026 10.1137/050647797 10.1016/j.apnum.2013.01.005 10.1137/060669073 10.1023/A:1012784129883 10.1080/01630563.2010.519136 10.1002/mma.3759 10.1016/j.enganabound.2009.07.008 10.1137/1.9781611972030 10.1016/j.camwa.2014.12.014 10.1016/j.apm.2011.08.032 10.1016/j.enganabound.2018.05.012 10.1016/0893-6080(94)90031-0 10.1007/s12206-009-0510-5 10.1016/j.camwa.2014.01.014 10.1016/S0893-6080(03)00083-2 10.1007/s10492-011-0024-1 10.5937/SPSUNP1802071T 10.18637/jss.v060.i03 10.1090/S0025-5718-2011-02481-6 10.1007/s10492-007-0013-6 10.1016/j.joems.2013.03.001 10.1016/j.camwa.2011.06.037 10.1007/s10092-005-0097-x 10.1109/ICPR.2000.906151 10.1007/s00521-014-1762-2 10.1007/s11075-017-0454-2 10.1137/110822931
ContentType	Journal Article
Copyright	Springer-Verlag London Ltd., part of Springer Nature 2020 Springer-Verlag London Ltd., part of Springer Nature 2020.
Copyright_xml	– notice: Springer-Verlag London Ltd., part of Springer Nature 2020 – notice: Springer-Verlag London Ltd., part of Springer Nature 2020.
DBID	AAYXX CITATION 8FE 8FG AFKRA ARAPS BENPR BGLVJ CCPQU DWQXO HCIFZ P5Z P62 PHGZM PHGZT PKEHL PQEST PQGLB PQQKQ PQUKI PRINS
DOI	10.1007/s00521-020-04810-0
DatabaseName	CrossRef ProQuest SciTech Collection ProQuest Technology Collection ProQuest Central UK/Ireland Advanced Technologies & Computer Science Collection ProQuest Central Technology Collection ProQuest One ProQuest Central Korea SciTech Premium Collection Advanced Technologies & Aerospace Database ProQuest Advanced Technologies & Aerospace Collection Proquest Central Premium ProQuest One Academic (New) ProQuest One Academic Middle East (New) ProQuest One Academic Eastern Edition (DO NOT USE) ProQuest One Applied & Life Sciences ProQuest One Academic (retired) ProQuest One Academic UKI Edition ProQuest Central China
DatabaseTitle	CrossRef Advanced Technologies & Aerospace Collection Technology Collection ProQuest One Academic Middle East (New) ProQuest Advanced Technologies & Aerospace Collection ProQuest One Academic Eastern Edition SciTech Premium Collection ProQuest One Community College ProQuest Technology Collection ProQuest SciTech Collection ProQuest Central China ProQuest Central Advanced Technologies & Aerospace Database ProQuest One Applied & Life Sciences ProQuest One Academic UKI Edition ProQuest Central Korea ProQuest Central (New) ProQuest One Academic ProQuest One Academic (New)
DatabaseTitleList	Advanced Technologies & Aerospace Collection
Database_xml	– sequence: 1 dbid: P5Z name: Advanced Technologies & Aerospace Database url: https://search.proquest.com/hightechjournals sourceTypes: Aggregation Database
DeliveryMethod	fulltext_linktorsrc
Discipline	Computer Science
EISSN	1433-3058
EndPage	14171
ExternalDocumentID	10_1007_s00521_020_04810_0
GroupedDBID	-4Z -59 -5G -BR -EM -Y2 -~C .4S .86 .DC .VR 06D 0R~ 0VY 123 1N0 1SB 2.D 203 28- 29N 2J2 2JN 2JY 2KG 2LR 2P1 2VQ 2~H 30V 4.4 406 408 409 40D 40E 53G 5QI 5VS 67Z 6NX 8FE 8FG 8TC 8UJ 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJBT AAJKR AANZL AAOBN AARHV AARTL AASML AATNV AATVU AAUYE AAWCG AAYIU AAYQN AAYTO AAYZH ABAKF ABBBX ABBXA ABDBF ABDZT ABECU ABFTD ABFTV ABHLI ABHQN ABJNI ABJOX ABKCH ABKTR ABLJU ABMNI ABMQK ABNWP ABQBU ABQSL ABSXP ABTEG ABTHY ABTKH ABTMW ABULA ABWNU ABXPI ACAOD ACBXY ACDTI ACGFS ACHSB ACHXU ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACSNA ACUHS ACZOJ ADHHG ADHIR ADIMF ADINQ ADKNI ADKPE ADMLS ADRFC ADTPH ADURQ ADYFF ADZKW AEBTG AEFIE AEFQL AEGAL AEGNC AEJHL AEJRE AEKMD AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AFBBN AFEXP AFGCZ AFKRA AFLOW AFQWF AFWTZ AFZKB AGAYW AGDGC AGGDS AGJBK AGMZJ AGQEE AGQMX AGRTI AGWIL AGWZB AGYKE AHAVH AHBYD AHKAY AHSBF AHYZX AIAKS AIGIU AIIXL AILAN AITGF AJBLW AJRNO AJZVZ ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARAPS ARCSS ARMRJ ASPBG AVWKF AXYYD AYJHY AZFZN B-. B0M BA0 BBWZM BDATZ BENPR BGLVJ BGNMA BSONS CAG CCPQU COF CS3 CSCUP DDRTE DL5 DNIVK DPUIP DU5 EAD EAP EBLON EBS ECS EDO EIOEI EJD EMI EMK EPL ESBYG EST ESX F5P FEDTE FERAY FFXSO FIGPU FINBP FNLPD FRRFC FSGXE FWDCC GGCAI GGRSB GJIRD GNWQR GQ6 GQ7 GQ8 GXS H13 HCIFZ HF~ HG5 HG6 HMJXF HQYDN HRMNR HVGLF HZ~ I-F I09 IHE IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C J0Z JBSCW JCJTX JZLTJ KDC KOV KOW LAS LLZTM M4Y MA- N2Q N9A NB0 NDZJH NPVJJ NQJWS NU0 O9- O93 O9G O9I O9J OAM P19 P2P P62 P9O PF0 PT4 PT5 QOK QOS R4E R89 R9I RHV RIG RNI RNS ROL RPX RSV RZK S16 S1Z S26 S27 S28 S3B SAP SCJ SCLPG SCO SDH SDM SHX SISQX SJYHP SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 T16 TSG TSK TSV TUC TUS U2A UG4 UOJIU UTJUX UZXMN VC2 VFIZW W23 W48 WK8 YLTOR Z45 Z5O Z7R Z7S Z7V Z7W Z7X Z7Y Z7Z Z81 Z83 Z86 Z88 Z8M Z8N Z8P Z8Q Z8R Z8S Z8T Z8U Z8W Z92 ZMTXR ~8M ~EX AAPKM AAYXX ABBRH ABDBE ABFSG ABRTQ ACSTC ADHKG ADKFA AEZWR AFDZB AFFHD AFHIU AFOHR AGQPQ AHPBZ AHWEU AIXLP ATHPR AYFIA CITATION PHGZM PHGZT PQGLB DWQXO PKEHL PQEST PQQKQ PQUKI PRINS
ID	FETCH-LOGICAL-c319t-aa12965e8209d3ec74a58d9b4a19678850f8c6d49543050124aeff98a856e70f3
IEDL.DBID	P5Z
ISICitedReferencesCount	11
ISICitedReferencesURI	http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000560557000069&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN	0941-0643
IngestDate	Wed Nov 05 02:13:40 EST 2025 Sat Nov 29 02:59:15 EST 2025 Tue Nov 18 21:56:24 EST 2025 Fri Feb 21 02:35:49 EST 2025
IsPeerReviewed	true
IsScholarly	true
Issue	17
Keywords	RBF neural network Nonlocal nonlinear problem Two-point step size gradient method Variable learning rate
Language	English
LinkModel	DirectLink
MergedId	FETCHMERGED-LOGICAL-c319t-aa12965e8209d3ec74a58d9b4a19678850f8c6d49543050124aeff98a856e70f3
Notes	ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14
PQID	2434612174
PQPubID	2043988
PageCount	19
ParticipantIDs	proquest_journals_2434612174 crossref_primary_10_1007_s00521_020_04810_0 crossref_citationtrail_10_1007_s00521_020_04810_0 springer_journals_10_1007_s00521_020_04810_0
PublicationCentury	2000
PublicationDate	20200900 2020-09-00 20200901
PublicationDateYYYYMMDD	2020-09-01
PublicationDate_xml	– month: 9 year: 2020 text: 20200900
PublicationDecade	2020
PublicationPlace	London
PublicationPlace_xml	– name: London – name: Heidelberg
PublicationTitle	Neural computing & applications
PublicationTitleAbbrev	Neural Comput & Applic
PublicationYear	2020
Publisher	Springer London Springer Nature B.V
Publisher_xml	– name: Springer London – name: Springer Nature B.V
References	Davydov, Oanhb (CR35) 2011; 62 Brandts, Korotov, Křížek, Šolc (CR5) 2009; 51 Grisvard (CR17) 2011 Themistoclakis, Vecchio (CR22) 2016; 292 Todorov (CR50) 2013; 66 Dai (CR24) 2012; 58 Kumar, Singh, Srivastava (CR42) 2013; 21 Andreev, Petrov, Todorov (CR49) 2005; 42 Petrov, Todorov (CR51) 2018; 79 Kanca (CR29) 2016; 39 Jianyu, Siwei, Yingjian, Yaping (CR12) 2003; 16 Khosravifard, Hematiyan (CR46) 2010; 34 Kumar, Kumar (CR25) 2009; 40 CR8 CR7 Szabó, Babuška (CR45) 1991 Ilati, Dehghan (CR11) 2018; 327 Aksoylu, Mengesha (CR19) 2020; 31 Brandts, Korotov, Křížek (CR4) 2007; 52 Gudi (CR26) 2012; 50 Liu, Wang (CR34) 2018; 100 Gaspar, Lisbona, Rodrigo, Kreiss, Lutstedt, Melqvist, Neytcheva (CR1) 2010 Wu (CR41) 1995; 4 Nie, Shu, Yu, An (CR30) 2014; 227 Kazem, Rad (CR37) 2012; 36 Sajavičius (CR33) 2014; 67 Li, Mazzucato, Nistor (CR48) 2010; 37 Hammer, Marlowe, Stroud (CR47) 1956; 10 Acosta, Armentano, Durán, Lombardi (CR15) 2007; 45 Dehghan (CR21) 2002; 19 Todorov (CR39) 2018; 76 Demlow, Georgoulis (CR18) 2012; 50 De Schepper, Slodivcka (CR32) 2000; 42 Corrêa, Menezes (CR20) 2004; 2004 Kim, Kim (CR38) 2009; 23 Birgin, Martínez, Raydan (CR43) 2014; 60 Tsoulos, Gavrilis, Glavas (CR14) 2009; 72 Todorov (CR40) 2015; 69 Jianyua, Siweia, Yingjiana, Yapinga (CR6) 2003; 16 Alves, Correa, Figueiredo (CR23) 2010; 2 Acosta, Armentano (CR16) 2011; 80 Nie, Yu (CR31) 2013; 68 Atluri, Shen (CR53) 2002; 3 Sanni (CR27) 2014; 2014 Aarts, van der Veer (CR13) 2001; 14 Shirvany, Hayati, Moradian (CR10) 2008; 13 Jung, Todorov (CR2) 2006; 56 Liu, Hong, Atluri (CR44) 2010; 60 Takeuchi, Kosugi (CR9) 1994; 7 Brandts, Korotov, Křížek (CR3) 2011; 56 Amster, De Napoli (CR28) 2006; 2006 Baymani, Effati, Niazmand, Kerayechian (CR36) 2015; 26 Todorov (CR52) 2018; 10 SN Atluri (4810_CR53) 2002; 3 4810_CR7 4810_CR8 L Jianyu (4810_CR12) 2003; 16 H Li (4810_CR48) 2010; 37 J Takeuchi (4810_CR9) 1994; 7 AB Andreev (4810_CR49) 2005; 42 S Kazem (4810_CR37) 2012; 36 B Szabó (4810_CR45) 1991 B Aksoylu (4810_CR19) 2020; 31 M Dehghan (4810_CR21) 2002; 19 W Themistoclakis (4810_CR22) 2016; 292 C Nie (4810_CR30) 2014; 227 G Dai (4810_CR24) 2012; 58 O Davydov (4810_CR35) 2011; 62 EG Birgin (4810_CR43) 2014; 60 FJS Corrêa (4810_CR20) 2004; 2004 M Kumar (4810_CR25) 2009; 40 Y Shirvany (4810_CR10) 2008; 13 J Brandts (4810_CR4) 2007; 52 M Ilati (4810_CR11) 2018; 327 CO Alves (4810_CR23) 2010; 2 G Acosta (4810_CR15) 2007; 45 M Baymani (4810_CR36) 2015; 26 G Acosta (4810_CR16) 2011; 80 IG Tsoulos (4810_CR14) 2009; 72 Miroslav S Petrov (4810_CR51) 2018; 79 J Brandts (4810_CR3) 2011; 56 P Amster (4810_CR28) 2006; 2006 TD Todorov (4810_CR52) 2018; 10 TD Todorov (4810_CR39) 2018; 76 C Nie (4810_CR31) 2013; 68 PC Hammer (4810_CR47) 1956; 10 L Jianyua (4810_CR6) 2003; 16 J Brandts (4810_CR5) 2009; 51 W Liu (4810_CR34) 2018; 100 H De Schepper (4810_CR32) 2000; 42 M Kumar (4810_CR42) 2013; 21 S Sajavičius (4810_CR33) 2014; 67 Z Wu (4810_CR41) 1995; 4 T Gudi (4810_CR26) 2012; 50 TD Todorov (4810_CR40) 2015; 69 P Grisvard (4810_CR17) 2011 TD Todorov (4810_CR50) 2013; 66 A Khosravifard (4810_CR46) 2010; 34 LP Aarts (4810_CR13) 2001; 14 SA Sanni (4810_CR27) 2014; 2014 Chein-Shan Liu (4810_CR44) 2010; 60 F Kanca (4810_CR29) 2016; 39 FJ Gaspar (4810_CR1) 2010 M Jung (4810_CR2) 2006; 56 A Demlow (4810_CR18) 2012; 50 Hyun-Gyu Kim (4810_CR38) 2009; 23
References_xml	– volume: 58 start-page: 1 year: 2012 end-page: 12 ident: CR24 article-title: On positive solutions for a class of nonlocal problems publication-title: Electron J Qual Theory Differ Equ – volume: 42 start-page: 518 year: 2000 end-page: 535 ident: CR32 article-title: Recovery of the boundary data for a linear second order elliptic problem with a nonlocal boundary condition publication-title: ANZIAM J – volume: 292 start-page: 720 issue: 15 year: 2016 end-page: 731 ident: CR22 article-title: On the numerical solution of a nonlocal boundary value problem publication-title: J Comput Appl Math – volume: 68 start-page: 31 year: 2013 end-page: 38 ident: CR31 article-title: Some error estimates on the finite element approximation for two-dimensional elliptic problem with nonlocal boundary publication-title: Appl Numer Math – volume: 80 start-page: 1949 issue: 276 year: 2011 end-page: 1978 ident: CR16 article-title: Finite element approximations in a non-Lipschitz domain: part II publication-title: Math Comput – volume: 50 start-page: 2159 issue: 5 year: 2012 end-page: 2181 ident: CR18 article-title: Pointwise a posteriori error control for discontinuous Galerkin methods for elliptic problems publication-title: SIAM J Numer Anal – volume: 39 start-page: 3152 issue: 11 year: 2016 end-page: 3158 ident: CR29 article-title: Inverse coefficient problem for a second-order elliptic equation with nonlocal boundary conditions publication-title: Math Methods Appl Sci – ident: CR8 – volume: 72 start-page: 2385 issue: 10–12 year: 2009 end-page: 2391 ident: CR14 article-title: Solving differential equations with constructed neural networks publication-title: Neurocomputing – volume: 21 start-page: 334 year: 2013 end-page: 339 ident: CR42 article-title: Various Newton-type iterative methods for solving nonlinear equations publication-title: J Egypt Math Soc – volume: 37 start-page: 41 year: 2010 end-page: 69 ident: CR48 article-title: Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains publication-title: Electron Trans Numer Anal – volume: 10 start-page: 130 year: 1956 end-page: 137 ident: CR47 article-title: Numerical integration over simplexes and cones publication-title: Math Tables Aids Comput – volume: 23 start-page: 2224 year: 2009 end-page: 2235 ident: CR38 article-title: An adaptive procedure based on combining finite elements with meshless methods publication-title: J Mech Sci Technol – volume: 100 start-page: 256 year: 2018 end-page: 264 ident: CR34 article-title: A modified approximate method based on Gaussian radial basis functions publication-title: Eng Anal Bound Elem – volume: 2004 start-page: 1 issue: 19 year: 2004 end-page: 10 ident: CR20 article-title: Existence of solutions to nonlocal and singular elliptic problems via Galerkin method publication-title: Electron J Differ Equ – volume: 26 start-page: 765 issue: 4 year: 2015 end-page: 773 ident: CR36 article-title: Artificial neural network method for solving the Navier–Stokes equations publication-title: Neural Comput Appl – volume: 16 start-page: 729 year: 2003 end-page: 734 ident: CR6 article-title: Numerical solution of elliptic partial differential equation using radial basis function neural networks publication-title: Neural Netw – volume: 7 start-page: 389 issue: 2 year: 1994 end-page: 395 ident: CR9 article-title: Neural network representation of finite element method publication-title: Neural Netw – volume: 66 start-page: 1272 issue: 7 year: 2013 end-page: 1283 ident: CR50 article-title: The optimal refinement strategies for 3-D simplicial meshes publication-title: Comput Math Appl – volume: 60 start-page: 1 issue: 3 year: 2014 end-page: 21 ident: CR43 article-title: Spectral projected gradient methods: review and perspectives publication-title: J Stat Softw – volume: 56 start-page: 417 year: 2011 end-page: 424 ident: CR3 article-title: Generalization of the Zlamal condition for simplicial finite elements in publication-title: Appl Math – volume: 19 start-page: 1 issue: 1 year: 2002 end-page: 12 ident: CR21 article-title: Numerical solution of a non-local boundary value problem with Neumann’s boundary conditions publication-title: Commun Numer Methods Eng – volume: 10 start-page: 71 issue: 2 year: 2018 end-page: 74 ident: CR52 article-title: A second order problem with nonlocal boundary conditions publication-title: Sci Publ State Univ Novi Pazar Ser A Appl Math Inform Mech – volume: 56 start-page: 1570 year: 2006 end-page: 1583 ident: CR2 article-title: Isoparametric multigrid method for reaction–diffusion equations publication-title: Appl Numer Math – volume: 2014 start-page: 1 issue: 124 year: 2014 end-page: 27 ident: CR27 article-title: Nonlocal degenerate reaction–diffusion equations with general nonlinear diffusion term publication-title: Electron J Differ Equ – volume: 2 start-page: 409 year: 2010 end-page: 417 ident: CR23 article-title: On a class of nonlocal elliptic problems with critical growth publication-title: Differ Equ Appl – volume: 79 start-page: 633 issue: 2 year: 2018 end-page: 656 ident: CR51 article-title: Stable subdivision of 4D polytopes publication-title: Numer Algorithms – volume: 36 start-page: 2360 year: 2012 end-page: 2369 ident: CR37 article-title: Radial basis functions method for solving of a non-local boundary value problem with Neumann’s boundary conditions publication-title: Appl Math Model – volume: 51 start-page: 317 issue: 2 year: 2009 end-page: 335 ident: CR5 article-title: On nonobtuse simplicial partitions publication-title: SIAM Rev – volume: 67 start-page: 1407 year: 2014 end-page: 1420 ident: CR33 article-title: Radial basis function method for a multidimensional linear elliptic equation with nonlocal boundary conditions publication-title: Comput Math Appl – volume: 3 start-page: 11 issue: 1 year: 2002 end-page: 51 ident: CR53 article-title: The meshless local Petrov–Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods publication-title: Comput Model Eng Sci – volume: 16 start-page: 729 issue: 5–6 year: 2003 end-page: 734 ident: CR12 article-title: Numerical solution of elliptic partial differential equation using radial basis function neural networks publication-title: Neural Netw – volume: 76 start-page: 1261 issue: 6 year: 2018 end-page: 1274 ident: CR39 article-title: Dirichlet problem for a nonlocal -Laplacian elliptic equation publication-title: Comput Math Appl – volume: 40 start-page: 1104 year: 2009 end-page: 1111 ident: CR25 article-title: Computational method for finding various solutions for a quasilinear elliptic equation of Kirchhoff type publication-title: Adv Eng Softw – volume: 60 start-page: 279 issue: 3 year: 2010 end-page: 308 ident: CR44 article-title: Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems publication-title: Comput Model Eng Sci – volume: 45 start-page: 277 issue: 1 year: 2007 end-page: 295 ident: CR15 article-title: Finite element approximations in a non-Lipschitz domain publication-title: SIAM J Numer Anal – start-page: 425 year: 2011 ident: CR17 publication-title: Elliptic problems in nonsmooth domains – volume: 13 start-page: 2132 year: 2008 end-page: 2145 ident: CR10 article-title: Numerical solution of the nonlinear Schrödinger equation by feed forward neural networks publication-title: Commun Nonlinear Sci Numer Simul – volume: 327 start-page: 314 year: 2018 end-page: 324 ident: CR11 article-title: Error analysis of a meshless weak form method based on radial point interpolation technique for Sivashinsky equation arising in the alloy solidification problem publication-title: J Comput Appl Math – volume: 227 start-page: 212 year: 2014 end-page: 221 ident: CR30 article-title: A high order composite scheme for the second order elliptic problem with nonlocal boundary and its fast algorithm publication-title: Appl Math Comput – year: 1991 ident: CR45 publication-title: Finite element analysis – year: 2010 ident: CR1 article-title: Multigrid finite element method on semi-structured grids for the poroelasticity problem publication-title: Numerical mathematics and advanced applications – volume: 4 start-page: 283 year: 1995 end-page: 292 ident: CR41 article-title: Compactly supported positive definite radial functions publication-title: Adv Comput Math – volume: 62 start-page: 2143 year: 2011 end-page: 2161 ident: CR35 article-title: On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation publication-title: Comput Math Appl – volume: 2006 start-page: 1 year: 2006 end-page: 11 ident: CR28 article-title: A nonlinear second order problem with a nonlocal boundary condition publication-title: Abstr Appl Anal – volume: 42 start-page: 47 issue: 2 year: 2005 end-page: 69 ident: CR49 article-title: An optimal order numerical quadrature approximation of a planar isoparametric eigenvalue problem on triangular finite element meshes publication-title: Calcolo – volume: 14 start-page: 261 year: 2001 end-page: 271 ident: CR13 article-title: Neural network method for solving partial differential equations publication-title: Neural Process Lett – volume: 31 start-page: 1301 issue: 12 year: 2020 end-page: 1317 ident: CR19 article-title: Results on nonlocal boundary value problems publication-title: Numer Funct AnalOptim – ident: CR7 – volume: 52 start-page: 251 issue: 3 year: 2007 end-page: 265 ident: CR4 article-title: Simplicial finite elements in higher dimensions publication-title: Appl Math – volume: 34 start-page: 30 year: 2010 end-page: 40 ident: CR46 article-title: A new method for meshless integration in 2D and 3D Galerkin meshfree methods publication-title: Eng Anal Bound Elem – volume: 50 start-page: 657 year: 2012 end-page: 668 ident: CR26 article-title: Finite element method for a nonlocal problem of Kirchhoff type publication-title: SIAM J Numer Anal – volume: 69 start-page: 411 issue: 5 year: 2015 end-page: 422 ident: CR40 article-title: Nonlocal problem for a general second-order elliptic operator publication-title: Comput Math Appl – volume: 50 start-page: 2159 issue: 5 year: 2012 ident: 4810_CR18 publication-title: SIAM J Numer Anal doi: 10.1137/110846397 – volume: 56 start-page: 1570 year: 2006 ident: 4810_CR2 publication-title: Appl Numer Math doi: 10.1016/j.apnum.2005.11.007 – volume: 3 start-page: 11 issue: 1 year: 2002 ident: 4810_CR53 publication-title: Comput Model Eng Sci – volume: 4 start-page: 283 year: 1995 ident: 4810_CR41 publication-title: Adv Comput Math doi: 10.1007/BF03177517 – volume: 13 start-page: 2132 year: 2008 ident: 4810_CR10 publication-title: Commun Nonlinear Sci Numer Simul doi: 10.1016/j.cnsns.2007.04.024 – volume: 327 start-page: 314 year: 2018 ident: 4810_CR11 publication-title: J Comput Appl Math doi: 10.1016/j.cam.2017.06.022 – volume: 40 start-page: 1104 year: 2009 ident: 4810_CR25 publication-title: Adv Eng Softw doi: 10.1016/j.advengsoft.2009.06.003 – ident: 4810_CR7 – volume: 19 start-page: 1 issue: 1 year: 2002 ident: 4810_CR21 publication-title: Commun Numer Methods Eng doi: 10.1002/cnm.522 – volume: 76 start-page: 1261 issue: 6 year: 2018 ident: 4810_CR39 publication-title: Comput Math Appl doi: 10.1016/j.camwa.2018.06.019 – volume: 58 start-page: 1 year: 2012 ident: 4810_CR24 publication-title: Electron J Qual Theory Differ Equ doi: 10.14232/ejqtde.2012.1.58 – volume: 42 start-page: 518 year: 2000 ident: 4810_CR32 publication-title: ANZIAM J doi: 10.21914/anziamj.v42i0.611 – volume: 10 start-page: 130 year: 1956 ident: 4810_CR47 publication-title: Math Tables Aids Comput doi: 10.2307/2002483 – volume: 60 start-page: 279 issue: 3 year: 2010 ident: 4810_CR44 publication-title: Comput Model Eng Sci – volume: 72 start-page: 2385 issue: 10–12 year: 2009 ident: 4810_CR14 publication-title: Neurocomputing doi: 10.1016/j.neucom.2008.12.004 – volume-title: Finite element analysis year: 1991 ident: 4810_CR45 – volume: 292 start-page: 720 issue: 15 year: 2016 ident: 4810_CR22 publication-title: J Comput Appl Math doi: 10.1016/j.cam.2015.02.048 – volume: 2014 start-page: 1 issue: 124 year: 2014 ident: 4810_CR27 publication-title: Electron J Differ Equ – volume: 66 start-page: 1272 issue: 7 year: 2013 ident: 4810_CR50 publication-title: Comput Math Appl doi: 10.1016/j.camwa.2013.07.026 – volume: 45 start-page: 277 issue: 1 year: 2007 ident: 4810_CR15 publication-title: SIAM J Numer Anal doi: 10.1137/050647797 – volume: 2006 start-page: 1 year: 2006 ident: 4810_CR28 publication-title: Abstr Appl Anal – volume: 68 start-page: 31 year: 2013 ident: 4810_CR31 publication-title: Appl Numer Math doi: 10.1016/j.apnum.2013.01.005 – volume: 51 start-page: 317 issue: 2 year: 2009 ident: 4810_CR5 publication-title: SIAM Rev doi: 10.1137/060669073 – volume: 37 start-page: 41 year: 2010 ident: 4810_CR48 publication-title: Electron Trans Numer Anal – volume: 14 start-page: 261 year: 2001 ident: 4810_CR13 publication-title: Neural Process Lett doi: 10.1023/A:1012784129883 – volume: 31 start-page: 1301 issue: 12 year: 2020 ident: 4810_CR19 publication-title: Numer Funct AnalOptim doi: 10.1080/01630563.2010.519136 – volume: 39 start-page: 3152 issue: 11 year: 2016 ident: 4810_CR29 publication-title: Math Methods Appl Sci doi: 10.1002/mma.3759 – volume: 34 start-page: 30 year: 2010 ident: 4810_CR46 publication-title: Eng Anal Bound Elem doi: 10.1016/j.enganabound.2009.07.008 – start-page: 425 volume-title: Elliptic problems in nonsmooth domains year: 2011 ident: 4810_CR17 doi: 10.1137/1.9781611972030 – volume: 69 start-page: 411 issue: 5 year: 2015 ident: 4810_CR40 publication-title: Comput Math Appl doi: 10.1016/j.camwa.2014.12.014 – volume: 36 start-page: 2360 year: 2012 ident: 4810_CR37 publication-title: Appl Math Model doi: 10.1016/j.apm.2011.08.032 – volume: 100 start-page: 256 year: 2018 ident: 4810_CR34 publication-title: Eng Anal Bound Elem doi: 10.1016/j.enganabound.2018.05.012 – volume: 7 start-page: 389 issue: 2 year: 1994 ident: 4810_CR9 publication-title: Neural Netw doi: 10.1016/0893-6080(94)90031-0 – volume: 23 start-page: 2224 year: 2009 ident: 4810_CR38 publication-title: J Mech Sci Technol doi: 10.1007/s12206-009-0510-5 – volume: 67 start-page: 1407 year: 2014 ident: 4810_CR33 publication-title: Comput Math Appl doi: 10.1016/j.camwa.2014.01.014 – volume: 227 start-page: 212 year: 2014 ident: 4810_CR30 publication-title: Appl Math Comput – volume: 16 start-page: 729 issue: 5–6 year: 2003 ident: 4810_CR12 publication-title: Neural Netw doi: 10.1016/S0893-6080(03)00083-2 – volume: 56 start-page: 417 year: 2011 ident: 4810_CR3 publication-title: Appl Math doi: 10.1007/s10492-011-0024-1 – volume-title: Numerical mathematics and advanced applications year: 2010 ident: 4810_CR1 – volume: 10 start-page: 71 issue: 2 year: 2018 ident: 4810_CR52 publication-title: Sci Publ State Univ Novi Pazar Ser A Appl Math Inform Mech doi: 10.5937/SPSUNP1802071T – volume: 16 start-page: 729 year: 2003 ident: 4810_CR6 publication-title: Neural Netw doi: 10.1016/S0893-6080(03)00083-2 – volume: 60 start-page: 1 issue: 3 year: 2014 ident: 4810_CR43 publication-title: J Stat Softw doi: 10.18637/jss.v060.i03 – volume: 2 start-page: 409 year: 2010 ident: 4810_CR23 publication-title: Differ Equ Appl – volume: 80 start-page: 1949 issue: 276 year: 2011 ident: 4810_CR16 publication-title: Math Comput doi: 10.1090/S0025-5718-2011-02481-6 – volume: 52 start-page: 251 issue: 3 year: 2007 ident: 4810_CR4 publication-title: Appl Math doi: 10.1007/s10492-007-0013-6 – volume: 21 start-page: 334 year: 2013 ident: 4810_CR42 publication-title: J Egypt Math Soc doi: 10.1016/j.joems.2013.03.001 – volume: 62 start-page: 2143 year: 2011 ident: 4810_CR35 publication-title: Comput Math Appl doi: 10.1016/j.camwa.2011.06.037 – volume: 42 start-page: 47 issue: 2 year: 2005 ident: 4810_CR49 publication-title: Calcolo doi: 10.1007/s10092-005-0097-x – volume: 2004 start-page: 1 issue: 19 year: 2004 ident: 4810_CR20 publication-title: Electron J Differ Equ – ident: 4810_CR8 doi: 10.1109/ICPR.2000.906151 – volume: 26 start-page: 765 issue: 4 year: 2015 ident: 4810_CR36 publication-title: Neural Comput Appl doi: 10.1007/s00521-014-1762-2 – volume: 79 start-page: 633 issue: 2 year: 2018 ident: 4810_CR51 publication-title: Numer Algorithms doi: 10.1007/s11075-017-0454-2 – volume: 50 start-page: 657 year: 2012 ident: 4810_CR26 publication-title: SIAM J Numer Anal doi: 10.1137/110822931
SSID	ssj0004685
Score	2.288894
Snippet	This paper proposes a radial basis function (RBF) network-based method for solving a nonlinear second-order elliptic equation with Dirichlet boundary...
SourceID	proquest crossref springer
SourceType	Aggregation Database Enrichment Source Index Database Publisher
StartPage	14153
SubjectTerms	Artificial Intelligence Boundary conditions Boundary value problems Computational Biology/Bioinformatics Computational Science and Engineering Computer Science Data Mining and Knowledge Discovery Differential equations Dirichlet problem Divergence Finite element method Image Processing and Computer Vision Learning Mathematical analysis Methods Neural networks Numerical integration Original Article Probability and Statistics in Computer Science Radial basis function Variational methods
SummonAdditionalLinks	– databaseName: Springer Journals dbid: RSV link: http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV1LS8QwEB509eDF9Ymrq-TgTQPddtImR_GBB1kEH-ytpGkKglTZroL_3kk23VVRQc-dpmUemZlk5huAw0TaVJYp8kiYiGOBJVfKJLwUGNsYU4ysh8y_yoZDORqp69AU1rTV7u2VpN-pZ81u7gSTUt_YlSNKd4W7CEvCoc24HP3m_kM3pB_ESXmLq-nBJLTKfL_GZ3c0jzG_XIt6b3PR_d9_rsFqiC7ZyVQd1mHB1hvQbSc3sGDIm3DmMDmIsJ4WgbMWWZxRCMtIG90pA6ufau_pWOFHL43fmEMGtyzMoGm24O7i_Pb0kod5CtyQoU241uTcU2HJ6asysSZDLWSpCtRkhpQKi6iSJi0pZXI4YOS5UNuqUlJLkdosqpJt6NCn7Q4w19Az0FjEVhcoTKaMpECIor0k01VUFj0YtGzNTQAbdzMvHvMZTLJnU05syj2b8qgHR7N3nqdQG79S91tp5cHsmjzGBB0kWoY9OG6lM3_882q7fyPfg5XYC9jVmvWhMxm_2H1YNq-Th2Z84NXxHTyQ1jY priority: 102 providerName: Springer Nature
Title	Neural network approach for solving nonlocal boundary value problems
URI	https://link.springer.com/article/10.1007/s00521-020-04810-0 https://www.proquest.com/docview/2434612174
Volume	32
WOSCitedRecordID	wos000560557000069&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText	1
inHoldings	1
isFullTextHit
isPrint
journalDatabaseRights	– providerCode: PRVPQU databaseName: Advanced Technologies & Aerospace Database customDbUrl: eissn: 1433-3058 dateEnd: 20241214 omitProxy: false ssIdentifier: ssj0004685 issn: 0941-0643 databaseCode: P5Z dateStart: 20120101 isFulltext: true titleUrlDefault: https://search.proquest.com/hightechjournals providerName: ProQuest – providerCode: PRVPQU databaseName: Proquest Central customDbUrl: eissn: 1433-3058 dateEnd: 20241214 omitProxy: false ssIdentifier: ssj0004685 issn: 0941-0643 databaseCode: BENPR dateStart: 20120101 isFulltext: true titleUrlDefault: https://www.proquest.com/central providerName: ProQuest – providerCode: PRVAVX databaseName: Springer Journals customDbUrl: eissn: 1433-3058 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0004685 issn: 0941-0643 databaseCode: RSV dateStart: 19970101 isFulltext: true titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22 providerName: Springer Nature
link	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LT8MwDLZg48CF8RSDMeXADSpKm7bpCfHYxAFN03ho4lKlaSohoW6sA4l_j52lGyCxC5ce2vSh2o4dx_4-gGNf6FBkIXfcQLkOT3nmxLHynSzgnvZ4yF1tIPPvol5PDIdx3ybcSltWWc2JZqLORopy5Gce9zmhXUX8YvzmEGsU7a5aCo1VqBNKAlE39IPnb32RhpITVzBU3cN92zRjWucoH4pnPSpuFLQh_NMxLaLNXxukxu90G__94k3YsBEnu5ypyBas6GIbGhWbA7PGvQM3hNOBA4tZYTir0MYZhrUMNZQyD6wYFcb7sdTQMU0-GaGFa2Z5acpdeOx2Hq5vHcux4Cg0vqkjJTr8MNAYCMSZr1XEZSCyOOUSTROXx4GbCxVmuIwibDD0ZlzqPI-FFEGoIzf396CGr9b7wKjJ51zy1NMy5YGKYiUwOMII0I9k7mZpE86rH5woC0BOPBivyRw62QglQaEkRiiJ24ST-T3jGfzG0tGtShKJNcUyWYihCaeVLBeX_37awfKnHcK6Z9SH6s1aUJtO3vURrKmP6Us5aUP9qtPrD9pGIfE4uH_6AhQp4yo
linkProvider	ProQuest
linkToHtml	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V1LS8NAEB5qK-jFt1ife9CTBtNkk2wOIj4qLa2lSAVvcbPZgCCttlXxT_kbndkmrQr25sFrHpvHfDuP3ZlvAPZdoX2R-NyyPWVbPOaJFYbKtRKPO9rhPre1ocxvBq2WuLsL2wX4yGthKK0y14lGUSc9RWvkxw53ObFdBfz06dmirlG0u5q30BjBoqHf3zBkG5zUL1G-B45zVe1c1Kysq4ClEG5DS0o0cb6n0fSFiatVwKUnkjDmEsGIAaFnp0L5CQYOxIaF-ptLnaahkMLzdWCnLo47AyV6Ha8IpfNqq33zpRLTNAHFmInyibiblemYYj1agcWjDqVTCtqC_m4KJ_7tjy1ZY-muFv_bP1qChcynZmejSbAMBd1dgcW8XwXL1NcqXBITCV7YHaW-s5xPnaHjznAO0toK6_a6xr6z2DSc6r8z4kPXLOu8M1iD2z_5lnUo4qP1BjAqY6pIHjtaxtxTQagEun_o47qBTO0kLkMlF2ikMop16vTxGI3JoQ0IIgRBZEAQ2WU4HN_zNCIYmXr1di75KFM2g2gi9jIc5diZnP59tM3po-3BXK1z3Yya9VZjC-YdA13KrtuG4rD_ondgVr0OHwb93WwaMLj_a1R9AtJUPHc
linkToPdf	http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV3dS8NADA86RXxxfuJ06j34psWuTdvroziH4hjDL_ZWrndXEKSOrQr-9-Zu7TZFBfG56bXkgySX5BeAY5_rkKsQHTeQroMpKieOpe-oAD3tYYiutpD53ajX44NB3J-b4rfd7lVJcjLTYFCa8uJsqLKz6eCbuc2kNNgzrYnclHMXYQkpkzFNXbd3j3OTkXYpJ-Uwpr8H_XJs5vszPrumWbz5pURqPU-n_v9_Xoe1Mupk5xM12YAFnW9CvdrowEoD34K2weogwnzSHM4qxHFGoS0jLTW3Dyx_ya0HZKldyTR6ZwYxXLNyN814Gx46l_cXV065Z8GRZICFIwQ5_TDQFAzEytcyQhFwFacoyDwpRQ7cjMtQUSpl8MHIo6HQWRZzwYNQR27m70CNPq13gZlBn5bA1NMixUBGseQUIFEU6Ecic1XagFbF4kSWIORmF8ZzMoVPtmxKiE2JZVPiNuBk-s5wAsHxK3WzklxSmuM48dBHA5UWYQNOK0nNHv982t7fyI9gpd_uJN3r3s0-rHpW1qYdrQm1YvSqD2BZvhVP49Gh1dIPQ0Dh_g
openUrl	ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Neural+network+approach+for+solving+nonlocal+boundary+value+problems&rft.jtitle=Neural+computing+%26+applications&rft.au=Palade%2C+V&rft.au=Petrov%2C+M+S&rft.au=Todorov%2C+T+D&rft.date=2020-09-01&rft.pub=Springer+Nature+B.V&rft.issn=0941-0643&rft.eissn=1433-3058&rft.volume=32&rft.issue=17&rft.spage=14153&rft.epage=14171&rft_id=info:doi/10.1007%2Fs00521-020-04810-0
thumbnail_l	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0941-0643&client=summon
thumbnail_m	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0941-0643&client=summon
thumbnail_s	http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0941-0643&client=summon