{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T05:15:03Z","timestamp":1735622103222,"version":"3.32.0"},"reference-count":29,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T00:00:00Z","timestamp":1743465600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2025,4,1]],"date-time":"2025-04-01T00:00:00Z","timestamp":1743465600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2024,11,8]],"date-time":"2024-11-08T00:00:00Z","timestamp":1731024000000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100021856","name":"Ministero dell'Universit\u00e0 e della Ricerca","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100021856","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Mathematics and Computers in Simulation"],"published-print":{"date-parts":[[2025,4]]},"DOI":"10.1016\/j.matcom.2024.10.039","type":"journal-article","created":{"date-parts":[[2024,11,3]],"date-time":"2024-11-03T06:29:06Z","timestamp":1730615346000},"page":"80-93","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":0,"special_numbering":"C","title":["Investigating neural networks with groundwater flow equation loss"],"prefix":"10.1016","volume":"230","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7486-4918","authenticated-orcid":false,"given":"Vincenzo","family":"Schiano Di Cola","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0002-6190-2473","authenticated-orcid":false,"given":"Vittorio","family":"Bauduin","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8199-5772","authenticated-orcid":false,"given":"Marco","family":"Berardi","sequence":"additional","affiliation":[]},{"given":"Filippo","family":"Notarnicola","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4128-2588","authenticated-orcid":false,"given":"Salvatore","family":"Cuomo","sequence":"additional","affiliation":[]}],"member":"78","reference":[{"year":"2024","series-title":"What is groundwater?","author":"U.S. Geological Survey","key":"10.1016\/j.matcom.2024.10.039_b1"},{"issue":"1","key":"10.1016\/j.matcom.2024.10.039_b2","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1016\/S0045-7930(01)00102-5","article-title":"Coupling of free surface and groundwater flows","volume":"32","author":"Miglio","year":"2003","journal-title":"Comput. & Fluids"},{"key":"10.1016\/j.matcom.2024.10.039_b3","doi-asserted-by":"crossref","first-page":"469","DOI":"10.1007\/s11242-021-01730-y","article-title":"Optimizing water consumption in Richards\u2019 equation framework with step-wise root water uptake: a simplified model","volume":"142","author":"Berardi","year":"2022","journal-title":"Transp. Porous Media"},{"issue":"9","key":"10.1016\/j.matcom.2024.10.039_b4","doi-asserted-by":"crossref","first-page":"1883","DOI":"10.1016\/j.advwatres.2007.02.007","article-title":"Adaptive local discontinuous Galerkin approximation to richards\u2019 equation","volume":"30","author":"Li","year":"2007","journal-title":"Adv. Water Resour."},{"issue":"2","key":"10.1016\/j.matcom.2024.10.039_b5","doi-asserted-by":"crossref","first-page":"341","DOI":"10.1007\/s10596-016-9566-3","article-title":"A study on iterative methods for solving richards\u2019 equation","volume":"20","author":"List","year":"2016","journal-title":"Comput. Geosci."},{"key":"10.1016\/j.matcom.2024.10.039_b6","doi-asserted-by":"crossref","first-page":"1990","DOI":"10.1016\/j.camwa.2019.07.026","article-title":"A mixed MoL-TMoL for the numerical solution of the 2D Richards\u2019 equation in layered soils","volume":"79","author":"Berardi","year":"2020","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.matcom.2024.10.039_b7","doi-asserted-by":"crossref","DOI":"10.1016\/j.cnsns.2023.107583","article-title":"Modeling plant water deficit by a non-local root water uptake term in the unsaturated flow equation","volume":"128","author":"Berardi","year":"2024","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"10.1016\/j.matcom.2024.10.039_b8","doi-asserted-by":"crossref","DOI":"10.1016\/j.advwatres.2022.104243","article-title":"GW-PINN: A deep learning algorithm for solving groundwater flow equations","volume":"165","author":"Zhang","year":"2022","journal-title":"Adv. Water Resour."},{"issue":"4","key":"10.1016\/j.matcom.2024.10.039_b9","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 Systems"},{"issue":"1","key":"10.1016\/j.matcom.2024.10.039_b10","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/s00365-021-09551-4","article-title":"A theoretical analysis of deep neural networks and parametric PDEs","volume":"55","author":"Kutyniok","year":"2022","journal-title":"Constr. Approx."},{"key":"10.1016\/j.matcom.2024.10.039_b11","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/j.neunet.2017.07.002","article-title":"Error bounds for approximations with deep ReLU networks","volume":"94","author":"Yarotsky","year":"2017","journal-title":"Neural Netw."},{"key":"10.1016\/j.matcom.2024.10.039_b12","doi-asserted-by":"crossref","first-page":"732","DOI":"10.1016\/j.neunet.2021.08.015","article-title":"On the approximation of functions by tanh neural networks","volume":"143","author":"De Ryck","year":"2021","journal-title":"Neural Netw."},{"issue":"3","key":"10.1016\/j.matcom.2024.10.039_b13","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\u2019s next","volume":"92","author":"Cuomo","year":"2022","journal-title":"J. Sci. Comput."},{"key":"10.1016\/j.matcom.2024.10.039_b14","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1017\/S0962492923000089","article-title":"Numerical analysis of physics-informed neural networks and related models in physics-informed machine learning","volume":"33","author":"De Ryck","year":"2024","journal-title":"Acta Numer."},{"key":"10.1016\/j.matcom.2024.10.039_b15","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2019.109136","article-title":"Adaptive activation functions accelerate convergence in deep and physics-informed neural networks","volume":"404","author":"Jagtap","year":"2020","journal-title":"J. Comput. Phys."},{"issue":"1","key":"10.1016\/j.matcom.2024.10.039_b16","doi-asserted-by":"crossref","first-page":"63","DOI":"10.3390\/math12010063","article-title":"Physics-informed neural networks with periodic activation functions for solute transport in heterogeneous porous media","volume":"12","author":"Faroughi","year":"2024","journal-title":"Mathematics"},{"key":"10.1016\/j.matcom.2024.10.039_b17","doi-asserted-by":"crossref","DOI":"10.1007\/s00366-024-01957-5","article-title":"Physics informed neural networks for an inverse problem in peridynamic models","author":"Difonzo","year":"2024","journal-title":"Eng. Comput."},{"key":"10.1016\/j.matcom.2024.10.039_b18","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2021.110768","article-title":"When and why PINNs fail to train: A neural tangent kernel perspective","volume":"449","author":"Wang","year":"2022","journal-title":"J. Comput. Phys."},{"key":"10.1016\/j.matcom.2024.10.039_b19","series-title":"Intelligent Systems and Applications","first-page":"490","article-title":"Determining the number of hidden layers in neural network by using principal component analysis","author":"Ibnu Choldun R.","year":"2020"},{"issue":"6","key":"10.1016\/j.matcom.2024.10.039_b20","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1007\/s10444-022-09985-9","article-title":"Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs","volume":"48","author":"De Ryck","year":"2022","journal-title":"Adv. Comput. Math."},{"key":"10.1016\/j.matcom.2024.10.039_b21","doi-asserted-by":"crossref","DOI":"10.1016\/j.jhydrol.2020.125948","article-title":"General analytical solutions of groundwater flow toward multi-dimensional sources\/sinks in a confined aquifer with leakage and distributed recharge","volume":"594","author":"Wang","year":"2021","journal-title":"J. Hydrol."},{"issue":"1","key":"10.1016\/j.matcom.2024.10.039_b22","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1007\/s11783-008-0067-z","article-title":"Analytical solutions of three-dimensional contaminant transport in uniform flow field in porous media: A library","volume":"3","author":"Wang","year":"2009","journal-title":"Front. Environ. Sci. Eng. China"},{"year":"2020","series-title":"Deep Learning Architectures: A Mathematical Approach","author":"Calin","key":"10.1016\/j.matcom.2024.10.039_b23"},{"year":"1999","series-title":"Neural Network Learning: Theoretical Foundations","author":"Anthony","key":"10.1016\/j.matcom.2024.10.039_b24"},{"year":"2022","series-title":"Mathematical Aspects of Deep Learning","author":"Grohs","key":"10.1016\/j.matcom.2024.10.039_b25"},{"year":"2024","series-title":"Collocation strategies for optimized PINNs in groundwater flow modeling","author":"Bauduin","key":"10.1016\/j.matcom.2024.10.039_b26"},{"key":"10.1016\/j.matcom.2024.10.039_b27","doi-asserted-by":"crossref","DOI":"10.1016\/j.rinam.2021.100200","article-title":"Remarks on the numerical approximation of Dirac delta functions","volume":"12","author":"Schiano Di Cola","year":"2021","journal-title":"Results Appl. Math."},{"key":"10.1016\/j.matcom.2024.10.039_b28","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1016\/j.camwa.2023.05.036","article-title":"Solving groundwater flow equation using physics-informed neural networks","volume":"145","author":"Cuomo","year":"2023","journal-title":"Comput. Math. Appl."},{"key":"10.1016\/j.matcom.2024.10.039_b29","doi-asserted-by":"crossref","unstructured":"S. Wang, S. Sankaran, H. Wang, P. Perdikaris, An Expert\u2019s Guide to Training Physics-informed Neural Networks, Technical Report, 2023, http:\/\/dx.doi.org\/10.48550\/arXiv.2308.08468, URL: . [physics] type: article.","DOI":"10.1016\/j.cma.2024.116813"}],"container-title":["Mathematics and Computers in Simulation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0378475424004373?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0378475424004373?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2024,12,30]],"date-time":"2024-12-30T15:34:20Z","timestamp":1735572860000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0378475424004373"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4]]},"references-count":29,"alternative-id":["S0378475424004373"],"URL":"https:\/\/doi.org\/10.1016\/j.matcom.2024.10.039","relation":{},"ISSN":["0378-4754"],"issn-type":[{"type":"print","value":"0378-4754"}],"subject":[],"published":{"date-parts":[[2025,4]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Investigating neural networks with groundwater flow equation loss","name":"articletitle","label":"Article Title"},{"value":"Mathematics and Computers in Simulation","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.matcom.2024.10.039","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2024 The Authors. Published by Elsevier B.V. on behalf of International Association for Mathematics and Computers in Simulation (IMACS).","name":"copyright","label":"Copyright"}]}}