{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T08:59:21Z","timestamp":1774601961056,"version":"3.50.1"},"reference-count":18,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T00:00:00Z","timestamp":1676332800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Research, Development and Innovation Office"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We propose neural-network-based algorithms for the numerical solution of boundary-value problems for the Laplace equation. Such a numerical solution is inherently mesh-free, and in the approximation process, stochastic algorithms are employed. The chief challenge in the solution framework is to generate appropriate learning data in the absence of the solution. Our main idea was to use fundamental solutions for this purpose and make a link with the so-called method of fundamental solutions. In this way, beyond the classical boundary-value problems, Dirichlet-to-Neumann operators can also be approximated. This problem was investigated in detail. Moreover, for this complex problem, low-rank approximations were constructed. Such efficient solution algorithms can serve as a basis for computational electrical impedance tomography.<\/jats:p>","DOI":"10.3390\/a16020111","type":"journal-article","created":{"date-parts":[[2023,2,14]],"date-time":"2023-02-14T01:41:06Z","timestamp":1676338866000},"page":"111","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Learning Data for Neural-Network-Based Numerical Solution of PDEs: Application to Dirichlet-to-Neumann Problems"],"prefix":"10.3390","volume":"16","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7398-7318","authenticated-orcid":false,"given":"Ferenc","family":"Izs\u00e1k","sequence":"first","affiliation":[{"name":"Alfr\u00e9d R\u00e9nyi Institute of Mathematics, 1053 Budapest, Hungary"},{"name":"Department of Applied Analysis, E\u00f6tv\u00f6s Lor\u00e1nd University, 1117 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5008-8130","authenticated-orcid":false,"given":"Taki Eddine","family":"Djebbar","sequence":"additional","affiliation":[{"name":"Department of Applied Analysis, E\u00f6tv\u00f6s Lor\u00e1nd University, 1117 Budapest, Hungary"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"686","DOI":"10.1016\/j.jcp.2018.10.045","article-title":"Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations","volume":"378","author":"Raissi","year":"2019","journal-title":"J. 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