{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T01:10:11Z","timestamp":1732669811003,"version":"3.28.2"},"reference-count":47,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T00:00:00Z","timestamp":1730419200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T00:00:00Z","timestamp":1730419200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2025,8,26]],"date-time":"2025-08-26T00:00:00Z","timestamp":1756166400000},"content-version":"am","delay-in-days":298,"URL":"http:\/\/www.elsevier.com\/open-access\/userlicense\/1.0\/"},{"start":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T00:00:00Z","timestamp":1730419200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-017"},{"start":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T00:00:00Z","timestamp":1730419200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-037"},{"start":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T00:00:00Z","timestamp":1730419200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-012"},{"start":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T00:00:00Z","timestamp":1730419200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-029"},{"start":{"date-parts":[[2024,11,1]],"date-time":"2024-11-01T00:00:00Z","timestamp":1730419200000},"content-version":"stm-asf","delay-in-days":0,"URL":"https:\/\/doi.org\/10.15223\/policy-004"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000038","name":"Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100000038","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000002","name":"National Institutes of Health","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100000002","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Computers & Mathematics with Applications"],"published-print":{"date-parts":[[2024,11]]},"DOI":"10.1016\/j.camwa.2024.08.013","type":"journal-article","created":{"date-parts":[[2024,8,27]],"date-time":"2024-08-27T00:27:43Z","timestamp":1724718463000},"page":"31-42","update-policy":"http:\/\/dx.doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Solutions to elliptic and parabolic problems via finite difference based unsupervised small linear convolutional neural networks"],"prefix":"10.1016","volume":"174","author":[{"ORCID":"http:\/\/orcid.org\/0000-0003-0041-7239","authenticated-orcid":false,"given":"Adrian","family":"Celaya","sequence":"first","affiliation":[]},{"given":"Keegan","family":"Kirk","sequence":"additional","affiliation":[]},{"given":"David","family":"Fuentes","sequence":"additional","affiliation":[]},{"ORCID":"http:\/\/orcid.org\/0000-0003-4230-2528","authenticated-orcid":false,"given":"Beatrice","family":"Riviere","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"key":"10.1016\/j.camwa.2024.08.013_br0010","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","article-title":"Multilayer feedforward networks are universal approximators","volume":"2","author":"Hornik","year":"1989","journal-title":"Neural Netw."},{"key":"10.1016\/j.camwa.2024.08.013_br0020","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/0893-6080(89)90003-8","article-title":"On the approximate realization of continuous mappings by neural networks","volume":"2","author":"Funahashi","year":"1989","journal-title":"Neural Netw."},{"issue":"3","key":"10.1016\/j.camwa.2024.08.013_br0030","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1002\/cnm.1640100303","article-title":"Neural-network-based approximations for solving partial differential equations","volume":"10","author":"Dissanayake","year":"1994","journal-title":"Commun. Numer. Methods Eng."},{"issue":"5","key":"10.1016\/j.camwa.2024.08.013_br0040","doi-asserted-by":"crossref","first-page":"987","DOI":"10.1109\/72.712178","article-title":"Artificial neural networks for solving ordinary and partial differential equations","volume":"9","author":"Lagaris","year":"1998","journal-title":"IEEE Trans. Neural Netw."},{"author":"Raissi","key":"10.1016\/j.camwa.2024.08.013_br0050"},{"author":"Raissi","key":"10.1016\/j.camwa.2024.08.013_br0060"},{"issue":"16","key":"10.1016\/j.camwa.2024.08.013_br0070","doi-asserted-by":"crossref","first-page":"15233","DOI":"10.1007\/s11071-023-08654-w","article-title":"Enhancing pinns for solving pdes via adaptive collocation point movement and adaptive loss weighting","volume":"111","author":"Hou","year":"2023","journal-title":"Nonlinear Dyn."},{"key":"10.1016\/j.camwa.2024.08.013_br0080","doi-asserted-by":"crossref","DOI":"10.1016\/j.compbiomed.2023.107287","article-title":"Physics-informed neural networks (pinns) for 4d hemodynamics prediction: an investigation of optimal framework based on vascular morphology","volume":"164","author":"Zhang","year":"2023","journal-title":"Comput. Biol. Med."},{"issue":"9","key":"10.1016\/j.camwa.2024.08.013_br0090","doi-asserted-by":"crossref","first-page":"2285","DOI":"10.1109\/TMI.2022.3161653","article-title":"Physics-informed neural networks for brain hemodynamic predictions using medical imaging","volume":"41","author":"Sarabian","year":"2022","journal-title":"IEEE Trans. Med. Imaging"},{"key":"10.1016\/j.camwa.2024.08.013_br0100","article-title":"Lpt-qpn: a lightweight physics-informed transformer for quantitative precipitation nowcasting","author":"Li","year":"2023","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"10.1016\/j.camwa.2024.08.013_br0110","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111731","article-title":"Optimal control of pdes using physics-informed neural networks","volume":"473","author":"Mowlavi","year":"2023","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2024.08.013_br0120","series-title":"2023 62nd IEEE Conference on Decision and Control (CDC)","first-page":"6014","article-title":"A physics-informed neural networks framework to solve the infinite-horizon optimal control problem","author":"Fotiadis","year":"2023"},{"key":"10.1016\/j.camwa.2024.08.013_br0130","doi-asserted-by":"crossref","DOI":"10.1016\/j.cma.2022.114909","article-title":"Can-pinn: a fast physics-informed neural network based on coupled-automatic\u2013numerical differentiation method","volume":"395","author":"Chiu","year":"2022","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"10.1016\/j.camwa.2024.08.013_br0140","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2022.111868","article-title":"Das-pinns: a deep adaptive sampling method for solving high-dimensional partial differential equations","volume":"476","author":"Tang","year":"2023","journal-title":"J. Comput. Phys."},{"author":"Ramachandran","key":"10.1016\/j.camwa.2024.08.013_br0150"},{"issue":"3","key":"10.1016\/j.camwa.2024.08.013_br0160","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/s10915-022-01939-z","article-title":"Scientific machine learning through physics\u2013informed neural networks: where we are and what's next","volume":"92","author":"Cuomo","year":"2022","journal-title":"J. Sci. Comput."},{"issue":"6","key":"10.1016\/j.camwa.2024.08.013_br0170","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1038\/s42254-021-00314-5","article-title":"Physics-informed machine learning","volume":"3","author":"Karniadakis","year":"2021","journal-title":"Nat. Rev. Phys."},{"author":"Bonfanti","key":"10.1016\/j.camwa.2024.08.013_br0180"},{"key":"10.1016\/j.camwa.2024.08.013_br0190","doi-asserted-by":"crossref","first-page":"1339","DOI":"10.1016\/j.jcp.2018.08.029","article-title":"DGM: a deep learning algorithm for solving partial differential equations","volume":"375","author":"Sirignano","year":"2018","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.camwa.2024.08.013_br0200","first-page":"1","article-title":"The deep Ritz method: a deep learning-based numerical algorithm for solving variational problems","volume":"6","author":"Wang","year":"2018","journal-title":"Commun. Math. Sci."},{"key":"10.1016\/j.camwa.2024.08.013_br0210","first-page":"1","article-title":"Error estimate for the deep Ritz method with boundary penalty","volume":"145","author":"M\u00fcller","year":"2022","journal-title":"Mach. Learn. Res."},{"issue":"4","key":"10.1016\/j.camwa.2024.08.013_br0220","doi-asserted-by":"crossref","first-page":"1172","DOI":"10.1109\/TMI.2022.3224873","article-title":"Pocketnet: a smaller neural network for medical image analysis","volume":"42","author":"Celaya","year":"2022","journal-title":"IEEE Trans. Med. Imaging"},{"key":"10.1016\/j.camwa.2024.08.013_br0230","series-title":"2022 IEEE International Conference on Systems, Man, and Cybernetics (SMC)","article-title":"Phyics informed neural network using finite difference method","author":"Lim","year":"2022"},{"key":"10.1016\/j.camwa.2024.08.013_br0240","doi-asserted-by":"crossref","DOI":"10.1016\/j.engappai.2022.105516","article-title":"Physics-informed convolutional neural networks for temperature field prediction of heat source without labeled data","volume":"117","author":"Zhao","year":"2023","journal-title":"Eng. Appl. Artif. Intell."},{"key":"10.1016\/j.camwa.2024.08.013_br0250","series-title":"Medical Image Computing and Computer-Assisted Intervention\u2013MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III, vol. 18","first-page":"234","article-title":"U-net: convolutional networks for biomedical image segmentation","author":"Ronneberger","year":"2015"},{"key":"10.1016\/j.camwa.2024.08.013_br0260","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TGRS.2023.3273149","article-title":"Inversion of time-lapse surface gravity data for detection of 3-D CO2 plumes via deep learning","volume":"61","author":"Celaya","year":"2023","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"10.1016\/j.camwa.2024.08.013_br0270","doi-asserted-by":"crossref","DOI":"10.1016\/j.jappgeo.2021.104507","article-title":"Deep learning inversion of gravity data for detection of CO2 plumes in overlying aquifers","volume":"196","author":"Yang","year":"2022","journal-title":"J. Appl. Geophys."},{"key":"10.1016\/j.camwa.2024.08.013_br0280","first-page":"550","article-title":"Deep learning joint inversion of seismic and electromagnetic data for salt reconstruction","volume":"2020","author":"Sun","year":"2020","journal-title":"SEG Tech. Program Expand. Abstr."},{"key":"10.1016\/j.camwa.2024.08.013_br0290","series-title":"Medical Image Computing and Computer-Assisted Intervention\u2013MICCAI 2016: 19th International Conference, Athens, Greece, October 17-21, 2016, Proceedings, Part II, vol. 19","first-page":"424","article-title":"3D U-Net: learning dense volumetric segmentation from sparse annotation","author":"\u00c7i\u00e7ek","year":"2016"},{"issue":"2","key":"10.1016\/j.camwa.2024.08.013_br0300","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1038\/s41592-020-01008-z","article-title":"nnU-Net: a self-configuring method for deep learning-based biomedical image segmentation","volume":"18","author":"Isensee","year":"122020","journal-title":"Nat. Methods"},{"issue":"1","key":"10.1016\/j.camwa.2024.08.013_br0310","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1016\/S0898-1221(03)90086-1","article-title":"A posteriori error estimates for a discontinuous Galerkin method applied to elliptic problems, log number: R74","volume":"46","author":"Riviere","year":"2003","journal-title":"Comput. Math. Appl."},{"author":"Kingma","key":"10.1016\/j.camwa.2024.08.013_br0320"},{"author":"Chollet","key":"10.1016\/j.camwa.2024.08.013_br0330"},{"issue":"5","key":"10.1016\/j.camwa.2024.08.013_br0340","doi-asserted-by":"crossref","first-page":"359","DOI":"10.1016\/0893-6080(89)90020-8","article-title":"Multilayer feedforward networks are universal approximators","volume":"2","author":"Hornik","year":"1989","journal-title":"Neural Netw."},{"issue":"4","key":"10.1016\/j.camwa.2024.08.013_br0350","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF02551274","article-title":"Approximation by superpositions of a sigmoidal function","volume":"2","author":"Cybenko","year":"1989","journal-title":"Math. Control Signals Syst."},{"issue":"3","key":"10.1016\/j.camwa.2024.08.013_br0360","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1016\/0893-6080(89)90003-8","article-title":"On the approximate realization of continuous mappings by neural networks","volume":"2","author":"Funahashi","year":"1989","journal-title":"Neural Netw."},{"key":"10.1016\/j.camwa.2024.08.013_br0370","article-title":"Efficient brain tumor segmentation with lightweight separable spatial convolutional network","author":"Zhang","year":"2024","journal-title":"ACM Trans. Multimed. Comput. Commun. Appl."},{"key":"10.1016\/j.camwa.2024.08.013_br0380","series-title":"Neural Information Processing Systems Conference: LatinX in AI (LXAI) Research Workshop 2023, LXAI","article-title":"Fmg-net and w-net: multigrid inspired deep learning architectures for medical imaging segmentation","author":"Celaya","year":"2023"},{"key":"10.1016\/j.camwa.2024.08.013_br0390","series-title":"Proceedings of the 2024 SIAM Conference on Parallel Processing for Scientific Computing (PP)","first-page":"1","article-title":"Inversion of time-lapse surface gravity data for monitoring of 3d co2 plumes via physics informed neural networks","author":"Celaya","year":"2024"},{"key":"10.1016\/j.camwa.2024.08.013_br0400","series-title":"CVPR","article-title":"Deep high-resolution representation learning for human pose estimation","author":"Sun","year":"2019"},{"key":"10.1016\/j.camwa.2024.08.013_br0410","series-title":"European Conference on Computer Vision","first-page":"630","article-title":"Identity mappings in deep residual networks","author":"He","year":"2016"},{"key":"10.1016\/j.camwa.2024.08.013_br0420","series-title":"Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition","first-page":"4700","article-title":"Densely connected convolutional networks","author":"Huang","year":"2017"},{"key":"10.1016\/j.camwa.2024.08.013_br0430","article-title":"Visualizing the loss landscape of neural nets","volume":"31","author":"Li","year":"2018","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"10.1016\/j.camwa.2024.08.013_br0440","unstructured":"C.-Y. Lee, S. Xie, P. Gallagher, Z. Zhang, Z. Tu, Deeply-supervised nets, in: G. Lebanon, S.V.N. Vishwanathan (Eds.), Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, San Diego, California, USA, 09\u201312 May 2015, in: Proceedings of Machine Learning Research, vol. 38, PMLR, pp. 562\u2013570."},{"author":"Li","key":"10.1016\/j.camwa.2024.08.013_br0450"},{"issue":"8","key":"10.1016\/j.camwa.2024.08.013_br0460","doi-asserted-by":"crossref","DOI":"10.1016\/j.heliyon.2023.e18820","article-title":"Enforcing Dirichlet boundary conditions in physics-informed neural networks and variational physics-informed neural networks","volume":"9","author":"Berrone","year":"2023","journal-title":"Heliyon"},{"author":"Zeinhofer","key":"10.1016\/j.camwa.2024.08.013_br0470"}],"container-title":["Computers & Mathematics with Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122124003663?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0898122124003663?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,11,27]],"date-time":"2024-11-27T00:49:30Z","timestamp":1732668570000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0898122124003663"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,11]]},"references-count":47,"alternative-id":["S0898122124003663"],"URL":"https:\/\/doi.org\/10.1016\/j.camwa.2024.08.013","relation":{},"ISSN":["0898-1221"],"issn-type":[{"type":"print","value":"0898-1221"}],"subject":[],"published":{"date-parts":[[2024,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Solutions to elliptic and parabolic problems via finite difference based unsupervised small linear convolutional neural networks","name":"articletitle","label":"Article Title"},{"value":"Computers & Mathematics with Applications","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.camwa.2024.08.013","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2024 Elsevier Ltd. All rights are reserved, including those for text and data mining, AI training, and similar technologies.","name":"copyright","label":"Copyright"}]}}