Numerical solution of (2+1)-dimensional nonlinear sine-Gordon equation with variable coefficients by using an efficient deep learning approach
In this article, we present an efficient neural-network-based deep learning approach, physics-informed neural networks (PINNs) with regularization technique, to resolve (2+1)-dimensional nonlinear damped and undamped sine-Gordon problem with variable coefficients. We suggest a multi-objective cost f...
Gespeichert in:
| Veröffentlicht in: | Zeitschrift für angewandte Mathematik und Physik Jg. 76; H. 4; S. 134 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Heidelberg
Springer Nature B.V
01.08.2025
|
| Schlagworte: | |
| ISSN: | 0044-2275, 1420-9039 |
| Online-Zugang: | Volltext |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Abstract | In this article, we present an efficient neural-network-based deep learning approach, physics-informed neural networks (PINNs) with regularization technique, to resolve (2+1)-dimensional nonlinear damped and undamped sine-Gordon problem with variable coefficients. We suggest a multi-objective cost function that incorporates the beginning circumstances, the governing problem’s residual, and various boundary circumstances to directly incorporate physical information of the proposed initial boundary value problem into the learning process. We employed a multiple densely connected network called feed-forward deep neural networks. To further enhance the network’s robustness and adaptability capabilities, we apply two regularization approaches to the PINNs. We point out three illustrative examples to demonstrate our suggested method’s effectiveness, validity, and practical consequences. The results showed that regularized PINNs produce better outcomes than the usual PINNs. We assess the accuracy of the model based on the relative, absolute, and training errors through tables and graphs. The findings suggested that the proposed machine learning approach PINNs is efficient and accurate and can be applied to any variable coefficient problem without requiring any linearization, perturbation, or interpolation techniques and the inclusion of appropriate regularization strategies in the PINNs can improve its performance. Therefore, to solve the variable coefficient nonlinear sine-Gordon equation and other difficult nonlinear physical issues in a range of fields, the PINNs is an appropriate programming machine learning method that is both accurate and efficient. |
|---|---|
| AbstractList | In this article, we present an efficient neural-network-based deep learning approach, physics-informed neural networks (PINNs) with regularization technique, to resolve (2+1)-dimensional nonlinear damped and undamped sine-Gordon problem with variable coefficients. We suggest a multi-objective cost function that incorporates the beginning circumstances, the governing problem’s residual, and various boundary circumstances to directly incorporate physical information of the proposed initial boundary value problem into the learning process. We employed a multiple densely connected network called feed-forward deep neural networks. To further enhance the network’s robustness and adaptability capabilities, we apply two regularization approaches to the PINNs. We point out three illustrative examples to demonstrate our suggested method’s effectiveness, validity, and practical consequences. The results showed that regularized PINNs produce better outcomes than the usual PINNs. We assess the accuracy of the model based on the relative, absolute, and training errors through tables and graphs. The findings suggested that the proposed machine learning approach PINNs is efficient and accurate and can be applied to any variable coefficient problem without requiring any linearization, perturbation, or interpolation techniques and the inclusion of appropriate regularization strategies in the PINNs can improve its performance. Therefore, to solve the variable coefficient nonlinear sine-Gordon equation and other difficult nonlinear physical issues in a range of fields, the PINNs is an appropriate programming machine learning method that is both accurate and efficient. |
| Author | Deresse, Alemayehu Tamirie Dufera, Tamirat Temesgen |
| Author_xml | – sequence: 1 givenname: Alemayehu Tamirie surname: Deresse fullname: Deresse, Alemayehu Tamirie – sequence: 2 givenname: Tamirat Temesgen surname: Dufera fullname: Dufera, Tamirat Temesgen |
| BookMark | eNo1T8tKAzEUDVLBtvoDrgJuFInmOZNZStEqFN3ouiSZG5synUwnM4o_4TcbX4vLgfPkztCkjS0gdMroFaO0vE6UUiEI5er7mCLVAZoyySmpqKgmaEqplITzUh2hWUrbbC8ZFVP0-TjuoA_ONDjFZhxCbHH0-JxfsgtShx20KVNZzYNNaMH0OGUgy9jX2Qr70fxk3sOwwW-mD8Y2gF0E74ML0A4J2w885swrNtn_T-MaoMNN7mt_pK7ro3GbY3ToTZPg5A_n6OXu9nlxT1ZPy4fFzYp0TKuBMA219b42TlvrmXegrZMlh8K7kltdKONMLZWsgHNnS22Z9QUIrSrJikKKOTr77c2z-xHSsN7Gsc9_prXgqqiELisuvgCxuGuh |
| ContentType | Journal Article |
| Copyright | The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025. |
| Copyright_xml | – notice: The Author(s), under exclusive licence to Springer Nature Switzerland AG 2025. |
| DOI | 10.1007/s00033-025-02515-9 |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics Physics |
| EISSN | 1420-9039 |
| GroupedDBID | -~C -~X .86 .DC .VR 06D 0R~ 0VY 123 203 29R 29~ 2J2 2JN 2JY 2KG 2KM 2LR 2~H 30V 4.4 406 408 409 40D 40E 5VS 67Z 6NX 6TJ 78A 8UJ 95- 95. 95~ 96X AAAVM AABHQ AACDK AAHNG AAIAL AAJKR AANZL AAPKM AARTL AASML AATNV AATVU AAUYE AAYIU AAYQN AAYZH ABAKF ABBBX ABBRH ABDBE ABDZT ABECU ABFSG ABFTV ABHLI ABHQN ABJNI ABJOX ABKCH ABKTR ABLJU ABMNI ABMQK ABNWP ABQBU ABRTQ ABSXP ABTEG ABTHY ABTKH ABTMW ABWNU ABXPI ACAOD ACDTI ACGFS ACHSB ACHXU ACIWK ACKNC ACMDZ ACMLO ACOKC ACOMO ACPIV ACSTC ACZOJ ADHHG ADHIR ADKNI ADKPE ADTPH ADURQ ADYFF ADZKW AEFQL AEGAL AEGNC AEJHL AEJRE AEMSY AENEX AEOHA AEPYU AESKC AETLH AEVLU AEXYK AEZWR AFDZB AFHIU AFOHR AFQWF AFWTZ AFZKB AGDGC AGJBK AGMZJ AGQEE AGQMX AGWIL AGWZB AGYKE AHAVH AHBYD AHPBZ AHSBF AHWEU AHYZX AIAKS AIGIU AIIXL AILAN AITGF AIXLP AJRNO AJZVZ ALMA_UNASSIGNED_HOLDINGS ALWAN AMKLP AMXSW AMYLF AMYQR AOCGG ARMRJ ASPBG ATHPR AVWKF AXYYD AYFIA AYJHY AZFZN B-. BA0 BGNMA BSONS CSCUP DDRTE DL5 DNIVK DPUIP DU5 EBLON EBS EIOEI ESBYG FEDTE FERAY FFXSO FIGPU FNLPD FRRFC FWDCC GGCAI GGRSB GJIRD GNWQR GQ7 GQ8 GXS HF~ HG5 HG6 HMJXF HQYDN HRMNR HVGLF HZ~ IJ- IKXTQ ITM IWAJR IXC IZIGR IZQ I~X I~Z J-C J0Z JBSCW JCJTX JZLTJ KDC KOV LAS LLZTM M4Y MA- MBV N9A NB0 NPVJJ NQJWS NU0 O9- O93 O9G O9I O9J OAM P19 P2P P9P PF0 PQQKQ PT4 PT5 QOK QOS R89 R9I RHV RNS ROL RPX RSV S16 S1Z S27 S3B SAP SDH SDM SEG SHX SISQX SJYHP SNE SNPRN SNX SOHCF SOJ SPISZ SRMVM SSLCW STPWE SZN T13 TSG TSK TSV TUC U2A UG4 UOJIU UTJUX VC2 W23 W48 WK8 YLTOR Z45 ZMTXR ~EX |
| ID | FETCH-LOGICAL-p185t-18edbffdac8bbf1fce8bc472e6fc72b865acad4549e22cb78b1bf6e3859416643 |
| ISICitedReferencesCount | 2 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=001505173600008&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0044-2275 |
| IngestDate | Tue Oct 07 07:23:56 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-p185t-18edbffdac8bbf1fce8bc472e6fc72b865acad4549e22cb78b1bf6e3859416643 |
| Notes | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| PQID | 3256938792 |
| PQPubID | 2043593 |
| ParticipantIDs | proquest_journals_3256938792 |
| PublicationCentury | 2000 |
| PublicationDate | 20250801 |
| PublicationDateYYYYMMDD | 2025-08-01 |
| PublicationDate_xml | – month: 08 year: 2025 text: 20250801 day: 01 |
| PublicationDecade | 2020 |
| PublicationPlace | Heidelberg |
| PublicationPlace_xml | – name: Heidelberg |
| PublicationTitle | Zeitschrift für angewandte Mathematik und Physik |
| PublicationYear | 2025 |
| Publisher | Springer Nature B.V |
| Publisher_xml | – name: Springer Nature B.V |
| SSID | ssj0007103 |
| Score | 2.4029188 |
| Snippet | In this article, we present an efficient neural-network-based deep learning approach, physics-informed neural networks (PINNs) with regularization technique,... |
| SourceID | proquest |
| SourceType | Aggregation Database |
| StartPage | 134 |
| SubjectTerms | Approximation Artificial neural networks Boundary conditions Boundary value problems Cost function Deep learning Finite volume method Inverse problems Iterative methods Machine learning Neural networks Numerical analysis Parameter estimation Partial differential equations Physics Regularization |
| Title | Numerical solution of (2+1)-dimensional nonlinear sine-Gordon equation with variable coefficients by using an efficient deep learning approach |
| URI | https://www.proquest.com/docview/3256938792 |
| Volume | 76 |
| WOSCitedRecordID | wos001505173600008&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: PRVAVX databaseName: SpringerLINK Contemporary 1997-Present customDbUrl: eissn: 1420-9039 dateEnd: 99991231 omitProxy: false ssIdentifier: ssj0007103 issn: 0044-2275 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/eLvHCXMwtV1Lj9MwELbKAhIcECwgHgvyARAoyqp5NfaRfcGhFARZVHGpbGdSIrbdbpKW3T_BL-THMLbjbFElBAcuUeW0eXS-ON945psh5JmMU4nAQSdHpNyPhZQ-j2Hg43OXy4RFfVCmuv4wHY3YeMw_9Ho_nRZmdZLO5-z8nC_-q6lxDI2tpbP_YO7uoDiAn9HouEWz4_avDD9a2iDMiefOowkhEsnwebiH8xb3c13R31bj8Oa2VIaoPJ0A779BXxR_AGe2Arhdpl2hP20UVuoUTMUJo4pD3rqsrcTR64a9HGDhWlFMu4rl6xT4C5QNutRVWTReoeP0e_s6j3MK34WWFnjvXCHZb7pHr2nOU3ZyogMwxc5bac5MXMDXpZeJWVmVHUIPlgWY7kl2h2i8DGZQT1vRW7vEESZdgt3vS5w6f1tHVToJjp3S49gPQ9t-ZRfsLB6jT8z7tkqSm-Ztm5kWzvHanB3Y1dSNd4lNH9F1tiId6tZCdiR_Pr98c7psgdH7ydHxcDjJDsfZi8WZr3ua6dh_2-DlCrmK18d1wuHHT587noDcrs1_sDfQSrqMsHPjpBtUwfCf7Da51Tou9LUF3B3Sg_k2udkZS9Xb5Lqxlarvkh8dCKkDIT0t6MvQC16tw4928KNr8KMOflTDjzr40XX4UXlBDfyowO-7YarhRx38qIPfPXJ8dJjtv_Xbzh_-Avlj4wcMclkUuVBMyiIoFDCp4jQErUwLJRskQok8TmIOYahkymQgiwFELOHoYCDJvk-28AbgAaE60KgkHlDieC6ARWkUJACDHLm8Yv2HZMf9sZP2Ka4nEToCPGIpDx_9efdjcuMSsDtkq6mW8IRcU6umrKunxt6_AJu7nq8 |
| linkProvider | Springer Nature |
| 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=Numerical+solution+of+%282%2B1%29-dimensional+nonlinear+sine-Gordon+equation+with+variable+coefficients+by+using+an+efficient+deep+learning+approach&rft.jtitle=Zeitschrift+f%C3%BCr+angewandte+Mathematik+und+Physik&rft.au=Deresse%2C+Alemayehu+Tamirie&rft.au=Dufera%2C+Tamirat+Temesgen&rft.date=2025-08-01&rft.pub=Springer+Nature+B.V&rft.issn=0044-2275&rft.eissn=1420-9039&rft.volume=76&rft.issue=4&rft.spage=134&rft_id=info:doi/10.1007%2Fs00033-025-02515-9&rft.externalDBID=NO_FULL_TEXT |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0044-2275&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0044-2275&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0044-2275&client=summon |