{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,24]],"date-time":"2026-04-24T03:02:42Z","timestamp":1776999762312,"version":"3.51.4"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T00:00:00Z","timestamp":1698019200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T00:00:00Z","timestamp":1698019200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Evolving Systems"],"published-print":{"date-parts":[[2024,8]]},"DOI":"10.1007\/s12530-023-09544-7","type":"journal-article","created":{"date-parts":[[2023,10,23]],"date-time":"2023-10-23T17:01:36Z","timestamp":1698080496000},"page":"1137-1158","update-policy":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Adaptive partition of unity networks (APUNet): a localized deep learning method for solving PDEs"],"prefix":"10.1007","volume":"15","author":[{"given":"Idriss","family":"Barbara","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/2.zoppoz.workers.dev:443\/https\/orcid.org\/0000-0002-7761-6300","authenticated-orcid":false,"given":"Tawfik","family":"Masrour","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Mohammed","family":"Hadda","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2023,10,23]]},"reference":[{"key":"9544_CR1","unstructured":"Al-Aradi A, Correia A, Naiff D, Jardim G, Saporito Y (2018) Solving nonlinear and high-dimensional partial differential equations via deep learning. arXiv preprint arXiv:1811.08782"},{"key":"9544_CR2","first-page":"127287","volume":"430","author":"A Al-Aradi","year":"2022","unstructured":"Al-Aradi A, Correia A, Jardim G, Freitas Naiff D, Saporito Y (2022) Extensions of the deep Galerkin method. Appl Math Comput 430:127287","journal-title":"Appl Math Comput"},{"issue":"4","key":"9544_CR3","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1002\/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N","volume":"40","author":"I Babu\u0161ka","year":"1997","unstructured":"Babu\u0161ka I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Eng 40(4):727\u2013758","journal-title":"Int J Numer Methods Eng"},{"key":"9544_CR4","unstructured":"Beck C, Hutzenthaler M, Jentzen A, Kuckuck B (2020) An overview on deep learning-based approximation methods for partial differential equations. arXiv preprint arXiv:2012.12348"},{"issue":"2","key":"9544_CR5","doi-asserted-by":"publisher","first-page":"202100006","DOI":"10.1002\/gamm.202100006","volume":"44","author":"J Blechschmidt","year":"2021","unstructured":"Blechschmidt J, Ernst OG (2021) Three ways to solve partial differential equations with neural networks?a review. GAMM-Mitteilungen 44(2):202100006","journal-title":"GAMM-Mitteilungen"},{"issue":"2","key":"9544_CR6","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1137\/16M1080173","volume":"60","author":"L Bottou","year":"2018","unstructured":"Bottou L, Curtis FE, Nocedal J (2018) Optimization methods for large-scale machine learning. SIAM Rev 60(2):223\u2013311","journal-title":"SIAM Rev"},{"key":"9544_CR7","unstructured":"Carmona R, Lauri\u00e8re M (2019) Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games: Ii\u2013the finite horizon case. arXiv preprint arXiv:1908.01613"},{"issue":"3","key":"9544_CR8","doi-asserted-by":"publisher","first-page":"1455","DOI":"10.1137\/19M1274377","volume":"59","author":"R Carmona","year":"2021","unstructured":"Carmona R, Lauri\u00e8re M (2021) Convergence analysis of machine learning algorithms for the numerical solution of mean field control and games i: the ergodic case. SIAM J Numer Anal 59(3):1455\u20131485","journal-title":"SIAM J Numer Anal"},{"issue":"7","key":"9544_CR9","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1002\/nme.455","volume":"54","author":"A Carpinteri","year":"2002","unstructured":"Carpinteri A, Ferro G, Ventura G (2002) The partition of unity quadrature in meshless methods. Int J Numer Methods Eng 54(7):987\u20131006","journal-title":"Int J Numer Methods Eng"},{"key":"9544_CR10","unstructured":"Chen J, Du R, Li P, Lyu L (2019) Quasi-Monte Carlo sampling for machine-learning partial differential equations. arXiv preprint arXiv:1911.01612"},{"issue":"3","key":"9544_CR11","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1002\/cnm.1640100303","volume":"10","author":"M Dissanayake","year":"1994","unstructured":"Dissanayake M, Phan-Thien N (1994) Neural-network-based approximations for solving partial differential equations. Commun Numer Methods Eng 10(3):195\u2013201","journal-title":"Commun Numer Methods Eng"},{"key":"9544_CR12","unstructured":"Grohs P, Hornung F, Jentzen A, Von\u00a0Wurstemberger P (2018) A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations. arXiv preprint arXiv:1809.02362"},{"key":"9544_CR13","doi-asserted-by":"crossref","unstructured":"He K, Zhang X, Ren S, Sun J (2016) Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on computer vision and pattern recognition, pp 770\u2013778","DOI":"10.1109\/CVPR.2016.90"},{"issue":"8","key":"9544_CR14","doi-asserted-by":"publisher","first-page":"1735","DOI":"10.1162\/neco.1997.9.8.1735","volume":"9","author":"S Hochreiter","year":"1997","unstructured":"Hochreiter S, Schmidhuber J (1997) Long short-term memory. Neural Comput 9(8):1735\u20131780. https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1162\/neco.1997.9.8.1735","journal-title":"Neural Comput"},{"key":"9544_CR15","unstructured":"Kingma DP, Ba J (2014) Adam: a method for stochastic optimization. arXiv preprint arXiv:1412.6980"},{"issue":"5","key":"9544_CR16","doi-asserted-by":"publisher","first-page":"987","DOI":"10.1109\/72.712178","volume":"9","author":"IE Lagaris","year":"1998","unstructured":"Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9(5):987\u20131000","journal-title":"IEEE Trans Neural Netw"},{"issue":"5","key":"9544_CR17","doi-asserted-by":"publisher","first-page":"1041","DOI":"10.1109\/72.870037","volume":"11","author":"IE Lagaris","year":"2000","unstructured":"Lagaris IE, Likas AC, Papageorgiou DG (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Netw 11(5):1041\u20131049","journal-title":"IEEE Trans Neural Netw"},{"key":"9544_CR18","unstructured":"Liao Y, Ming P (2019) Deep Nitsche method: deep Ritz method with essential boundary conditions. arXiv preprint arXiv:1912.01309"},{"key":"9544_CR19","doi-asserted-by":"publisher","first-page":"110930","DOI":"10.1016\/j.jcp.2021.110930","volume":"452","author":"L Lyu","year":"2022","unstructured":"Lyu L, Zhang Z, Chen M, Chen J (2022) Mim: a deep mixed residual method for solving high-order partial differential equations. J Comput Phys 452:110930","journal-title":"J Comput Phys"},{"issue":"6","key":"9544_CR20","doi-asserted-by":"publisher","first-page":"348","DOI":"10.2322\/tjsass.64.348","volume":"64","author":"M Matsumoto","year":"2021","unstructured":"Matsumoto M (2021) Application of deep Galerkin method to solve compressible Navier-Stokes equations. Trans Jpn Soc Aeronaut Sp Sci 64(6):348\u2013357","journal-title":"Trans Jpn Soc Aeronaut Sp Sci"},{"key":"9544_CR21","doi-asserted-by":"crossref","unstructured":"Nitsche J (1971) \u00dcber ein variationsprinzip zur l\u00f6sung von dirichlet-problemen bei verwendung von teilr\u00e4umen, die keinen randbedingungen unterworfen sind. In: Abhandlungen aus dem Mathematischen Seminar der Universit\u00e4t Hamburg, vol 36. Springer, pp 9\u201315","DOI":"10.1007\/BF02995904"},{"issue":"5","key":"9544_CR22","doi-asserted-by":"publisher","first-page":"503","DOI":"10.1007\/s11633-017-1054-2","volume":"14","author":"T Poggio","year":"2017","unstructured":"Poggio T, Mhaskar H, Rosasco L, Miranda B, Liao Q (2017) Why and when can deep-but not shallow-networks avoid the curse of dimensionality: a review. Int J Autom Comput 14(5):503\u2013519","journal-title":"Int J Autom Comput"},{"key":"9544_CR23","unstructured":"Raissi M, Perdikaris P, Karniadakis GE (2017a) Physics informed deep learning (part i): Data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10561"},{"key":"9544_CR24","unstructured":"Raissi M, Perdikaris P, Karniadakis GE (2017b) Physics informed deep learning (part ii): data-driven solutions of nonlinear partial differential equations. arXiv preprint arXiv:1711.10566"},{"key":"9544_CR25","doi-asserted-by":"publisher","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","volume":"378","author":"M Raissi","year":"2019","unstructured":"Raissi M, Perdikaris P, Karniadakis GE (2019) Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J Comput Phys 378:686\u2013707","journal-title":"J Comput Phys"},{"issue":"3","key":"9544_CR26","doi-asserted-by":"publisher","first-page":"912","DOI":"10.1137\/20M1329597","volume":"12","author":"YF Saporito","year":"2021","unstructured":"Saporito YF, Zhang Z (2021) Path-dependent deep Galerkin method: a neural network approach to solve path-dependent partial differential equations. SIAM J Financ Math 12(3):912\u2013940","journal-title":"SIAM J Financ Math"},{"key":"9544_CR27","doi-asserted-by":"crossref","unstructured":"Shepard D (1968) A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 1968 23rd ACM National Conference, pp 517\u2013524","DOI":"10.1145\/800186.810616"},{"key":"9544_CR28","doi-asserted-by":"publisher","first-page":"1339","DOI":"10.1016\/j.jcp.2018.08.029","volume":"375","author":"J Sirignano","year":"2018","unstructured":"Sirignano J, Spiliopoulos K (2018) Dgm: a deep learning algorithm for solving partial differential equations. J Comput Phys 375:1339\u20131364","journal-title":"J Comput Phys"},{"key":"9544_CR29","doi-asserted-by":"crossref","unstructured":"Vergunova I, Vergunov V, Rosemann I (2021) Solving the coefficient inverse problem by the deep Galerkin method. In: 2021 11th International Conference on advanced computer information technologies (ACIT). IEEE, pp 65\u201370","DOI":"10.1109\/ACIT52158.2021.9548633"},{"issue":"1","key":"9544_CR30","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/BF02123482","volume":"4","author":"H Wendland","year":"1995","unstructured":"Wendland H (1995) Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree. Adv Comput Math 4(1):389\u2013396","journal-title":"Adv Comput Math"},{"key":"9544_CR31","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s40304-018-0127-z","volume":"6","author":"B Yu","year":"2018","unstructured":"Yu B (2018) The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems. Commun Math Stat 6:1\u201312","journal-title":"Commun Math Stat"}],"container-title":["Evolving Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/content\/pdf\/10.1007\/s12530-023-09544-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/article\/10.1007\/s12530-023-09544-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/content\/pdf\/10.1007\/s12530-023-09544-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,23]],"date-time":"2024-07-23T10:40:05Z","timestamp":1721731205000},"score":1,"resource":{"primary":{"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/link.springer.com\/10.1007\/s12530-023-09544-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,10,23]]},"references-count":31,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2024,8]]}},"alternative-id":["9544"],"URL":"https:\/\/2.zoppoz.workers.dev:443\/https\/doi.org\/10.1007\/s12530-023-09544-7","relation":{},"ISSN":["1868-6478","1868-6486"],"issn-type":[{"value":"1868-6478","type":"print"},{"value":"1868-6486","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,10,23]]},"assertion":[{"value":"8 March 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 September 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 October 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}