{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T18:41:53Z","timestamp":1773945713813,"version":"3.50.1"},"publisher-location":"Cham","reference-count":24,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783031857027","type":"print"},{"value":"9783031857034","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,1,1]],"date-time":"2025-01-01T00:00:00Z","timestamp":1735689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025]]},"DOI":"10.1007\/978-3-031-85703-4_12","type":"book-chapter","created":{"date-parts":[[2025,4,2]],"date-time":"2025-04-02T03:29:13Z","timestamp":1743564553000},"page":"174-189","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Comparison of\u00a0Multigrid and\u00a0Machine Learning-Based Poisson Solvers"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6963-9302","authenticated-orcid":false,"given":"Hadrien","family":"God\u00e9","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4142-7356","authenticated-orcid":false,"given":"Carola","family":"Kruse","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7749-8239","authenticated-orcid":false,"given":"Richard","family":"Angersbach","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6992-2690","authenticated-orcid":false,"given":"Harald","family":"K\u00f6stler","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9550-9077","authenticated-orcid":false,"given":"Micha\u00ebl","family":"Bauerheim","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8796-8599","authenticated-orcid":false,"given":"Ulrich","family":"R\u00fcde","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,4,1]]},"reference":[{"key":"12_CR1","unstructured":"Pytorch 2: Faster machine learning through dynamic python bytecode transformation and graph compilation (2024)"},{"key":"12_CR2","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsv.2021.116285","volume":"512","author":"A Alguacil","year":"2021","unstructured":"Alguacil, A., Bauerheim, M., Jacob, M.C., Moreau, S.: Predicting the propagation of acoustic waves using deep convolutional neural networks. J. Sound Vib. 512, 116285 (2021)","journal-title":"J. Sound Vib."},{"issue":"138","key":"12_CR3","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1090\/S0025-5718-1977-0431719-X","volume":"31","author":"A Brandt","year":"1977","unstructured":"Brandt, A.: Multi-level adaptive solutions to boundary-value problems. Math. Comput. 31(138), 333\u2013390 (1977)","journal-title":"Math. Comput."},{"key":"12_CR4","doi-asserted-by":"crossref","unstructured":"Brandt, A., Livne, O.E.: Multigrid Techniques: 1984 Guide with Applications to Fluid Dynamics, Revised Edition. SIAM (2011)","DOI":"10.1137\/1.9781611970753"},{"key":"12_CR5","doi-asserted-by":"crossref","unstructured":"Briggs, W., Henson, V., McCormick, S.: A Multigrid Tutorial, 2nd edn. (2000)","DOI":"10.1137\/1.9780898719505"},{"key":"12_CR6","unstructured":"Cheng, L., Illarramendi, E.A., Bogopolsky, G., Bauerheim, M., Cuenot, B.: Using neural networks to solve the 2d poisson equation for electric field computation in plasma fluid simulations. ArXiv arxiv:2109.13076 (2021). https:\/\/www.semanticscholar.org\/paper\/2da0491ba38ac18ef03aee950caac3a6f94ab38e"},{"key":"12_CR7","unstructured":"Dai, L., Du, X., Zhang, H., Tang, J.: Mgnet: learning correspondences via multiple graphs (2024)"},{"key":"12_CR8","unstructured":"Estivalezes, J.L., et al.: A phase inversion benchmark for multiscale multiphase flows (2021). http:\/\/arxiv.org\/abs\/1906.02655"},{"key":"12_CR9","doi-asserted-by":"publisher","unstructured":"Grossmann, T.G., Komorowska, U.J., Latz, J., Sch\u00f6nlieb, C.B.: Can physics-informed neural networks beat the finite element method? IMA J. Appl. Math. 89(1), 143\u2013174 (2024). https:\/\/doi.org\/10.1093\/imamat\/hxae011","DOI":"10.1093\/imamat\/hxae011"},{"key":"12_CR10","doi-asserted-by":"crossref","unstructured":"Hackbusch, W.: Multi-Grid Methods and Applications. Springer, Heidelberg (1985)","DOI":"10.1007\/978-3-662-02427-0"},{"key":"12_CR11","unstructured":"Ibeid, H., Olson, L., Gropp, W.: Fft, fmm, and multigrid on the road to exascale: performance challenges and opportunities (2018)"},{"key":"12_CR12","unstructured":"Illarramendi, E.A., Bauerheim, M., Nadal, A.G.: Embedding temporal error propagation on cnn for unsteady flow simulations (2021). https:\/\/www.semanticscholar.org\/paper\/b9cd03b7c6791fa6099d46bd463e7c3b21bc17ad"},{"issue":"4","key":"12_CR13","doi-asserted-by":"publisher","first-page":"A2448","DOI":"10.1137\/21M1397520","volume":"43","author":"MJ K\u00fchn","year":"2021","unstructured":"K\u00fchn, M.J., Kruse, C., R\u00fcde, U.: Energy-minimizing, symmetric discretizations for anisotropic meshes and energy functional extrapolation. SIAM J. Sci. Comput. 43(4), A2448\u2013A2473 (2021)","journal-title":"SIAM J. Sci. Comput."},{"issue":"1","key":"12_CR14","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10915-022-01802-1","volume":"91","author":"MJ K\u00fchn","year":"2022","unstructured":"K\u00fchn, M.J., Kruse, C., R\u00fcde, U.: Implicitly extrapolated geometric multigrid on disk-like domains for the gyrokinetic Poisson equation from fusion plasma applications. J. Sci. Comput. 91(1), 1\u201327 (2022)","journal-title":"J. Sci. Comput."},{"key":"12_CR15","doi-asserted-by":"publisher","first-page":"405","DOI":"10.1007\/978-3-030-47956-5_14","volume-title":"Software for Exascale Computing - SPPEXA 2016\u20132019","author":"C Lengauer","year":"2020","unstructured":"Lengauer, C., et al.: Exastencils: advanced multigrid solver generation. In: Bungartz, H.J., Reiz, S., Uekermann, B., Neumann, P., Nagel, W.E. (eds.) Software for Exascale Computing - SPPEXA 2016\u20132019, pp. 405\u2013452. Springer, Cham (2020)"},{"issue":"5","key":"12_CR16","first-page":"973","volume":"3","author":"B Lu","year":"2008","unstructured":"Lu, B., Zhou, Y., Holst, M., McCammon, J.: Recent progress in numerical methods for the poisson-boltzmann equation in biophysical applications. Commun. Comput. Phys. 3(5), 973\u20131009 (2008)","journal-title":"Commun. Comput. Phys."},{"key":"12_CR17","unstructured":"Sadiku, M.N.: Electromagnetics (2001)"},{"key":"12_CR18","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1007\/s10710-021-09412-w","volume":"22","author":"J Schmitt","year":"2021","unstructured":"Schmitt, J., Kuckuk, S., K\u00f6stler, H.: Evostencils: a grammar-based genetic programming approach for constructing efficient geometric multigrid methods. Genet. Program Evolvable Mach. 22, 511\u2013537 (2021)","journal-title":"Genet. Program Evolvable Mach."},{"key":"12_CR19","doi-asserted-by":"crossref","unstructured":"Singh, R., Lenka, T., Panda, D., Nguyen, H., Boukortt, N.E.I., Crupi, G.: Analytical modeling of i\u2013v characteristics using 2d poisson equations in aln\/-ga2o3 hemt. Mater. Sci. Semicond. Process. 145, 106627 (2022)","DOI":"10.1016\/j.mssp.2022.106627"},{"key":"12_CR20","doi-asserted-by":"crossref","unstructured":"St\u00fcben, K., Trottenberg, U.: Multigrid methods: fundamental algorithms, model problem analysis and applications. In: Multigrid Methods: Proceedings of the Conference Held at K\u00f6ln-Porz, 23\u201327 November 1981, pp. 1\u2013176. Springer, Heidelberg (1982)","DOI":"10.1007\/BFb0069928"},{"key":"12_CR21","unstructured":"Tompson, J., Schlachter, K., Sprechmann, P., Perlin, K.: Accelerating eulerian fluid simulation with convolutional networks. In: International Conference on Machine Learning, pp. 3424\u20133433. PMLR (2017)"},{"key":"12_CR22","unstructured":"Trottenberg, U., Oosterlee, C.W., Schuller, A.: Multigrid. Elsevier, Amsterdam (2000)"},{"key":"12_CR23","unstructured":"Wandel, N., Weinmann, M., Klein, R.: Learning incompressible fluid dynamics from scratch\u2013towards fast, differentiable fluid models that generalize. arXiv preprint arXiv:2006.08762 (2020)"},{"key":"12_CR24","doi-asserted-by":"publisher","DOI":"10.1017\/dce.2021.7","volume":"2","author":"AG \u00d6zbay","year":"2021","unstructured":"\u00d6zbay, A.G., Hamzehloo, A., Laizet, S., Tzirakis, P., Rizos, G., Schuller, B.: Poisson cnn: convolutional neural networks for the solution of the poisson equation on a cartesian mesh. Data-Centric Eng. 2, e6 (2021). https:\/\/doi.org\/10.1017\/dce.2021.7","journal-title":"Data-Centric Eng."}],"container-title":["Lecture Notes in Computer Science","Parallel Processing and Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-85703-4_12","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,4,2]],"date-time":"2025-04-02T03:29:21Z","timestamp":1743564561000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-85703-4_12"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"ISBN":["9783031857027","9783031857034"],"references-count":24,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-85703-4_12","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025]]},"assertion":[{"value":"1 April 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"PPAM","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Parallel Processing and Applied Mathematics","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Ostrava","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Czech Republic","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2024","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"9 September 2024","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"12 September 2024","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"15","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"ppam2024","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/ppam.edu.pl\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}