{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,7]],"date-time":"2026-06-07T20:58:17Z","timestamp":1780865897652,"version":"3.54.1"},"reference-count":42,"publisher":"Elsevier BV","license":[{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/tdm\/userlicense\/1.0\/"},{"start":{"date-parts":[[2026,11,1]],"date-time":"2026-11-01T00:00:00Z","timestamp":1793491200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.elsevier.com\/legal\/tdmrep-license"},{"start":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T00:00:00Z","timestamp":1780358400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["elsevier.com","sciencedirect.com"],"crossmark-restriction":true},"short-container-title":["Neural Networks"],"published-print":{"date-parts":[[2026,11]]},"DOI":"10.1016\/j.neunet.2026.109208","type":"journal-article","created":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T08:12:05Z","timestamp":1780474325000},"page":"109208","update-policy":"https:\/\/doi.org\/10.1016\/elsevier_cm_policy","source":"Crossref","is-referenced-by-count":1,"special_numbering":"C","title":["Symplectic convolutional neural networks"],"prefix":"10.1016","volume":"203","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7904-605X","authenticated-orcid":false,"given":"S\u00fcleyman","family":"Y\u0131ld\u0131z","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-9030-0708","authenticated-orcid":false,"given":"Konrad","family":"Janik","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3362-4103","authenticated-orcid":false,"given":"Peter","family":"Benner","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"78","reference":[{"issue":"6","key":"10.1016\/j.neunet.2026.109208_bib0001","doi-asserted-by":"crossref","first-page":"A2616","DOI":"10.1137\/17M1111991","article-title":"Structure preserving model reduction of parametric Hamiltonian systems","volume":"39","author":"Afkham","year":"2017","journal-title":"SIAM Journal on Scientific Computing"},{"key":"10.1016\/j.neunet.2026.109208_bib0002","doi-asserted-by":"crossref","DOI":"10.1016\/j.jcp.2025.113832","article-title":"Structure-preserving dimensionality reduction for learning Hamiltonian dynamics","volume":"528","author":"Baj\u0101rs","year":"2025","journal-title":"Journal of Computational Physics"},{"issue":"8","key":"10.1016\/j.neunet.2026.109208_bib0003","doi-asserted-by":"crossref","first-page":"1798","DOI":"10.1109\/TPAMI.2013.50","article-title":"Representation learning: A review and new perspectives","volume":"35","author":"Bengio","year":"2013","journal-title":"IEEE Transactions on Pattern Analysis and Machine Intelligence"},{"key":"10.1016\/j.neunet.2026.109208_sbref0004","series-title":"Model order reduction. Volume 1: System- and data-driven methods and algorithms","year":"2021"},{"key":"10.1016\/j.neunet.2026.109208_bib0005","series-title":"Model order reduction. Volume 2: Snapshot-based methods and algorithms","year":"2021"},{"key":"10.1016\/j.neunet.2026.109208_bib0006","series-title":"Model order reduction. Volume 3: Applications","year":"2021"},{"issue":"1","key":"10.1016\/j.neunet.2026.109208_bib0007","doi-asserted-by":"crossref","first-page":"539","DOI":"10.1146\/annurev.fl.25.010193.002543","article-title":"The proper orthogonal decomposition in the analysis of turbulent flows","volume":"25","author":"Berkooz","year":"1993","journal-title":"Annual Review of Fluid Mechanics"},{"issue":"12","key":"10.1016\/j.neunet.2026.109208_bib0008","doi-asserted-by":"crossref","DOI":"10.1063\/1.5128231","article-title":"On learning Hamiltonian systems from data","volume":"29","author":"Bertalan","year":"2019","journal-title":"Chaos"},{"key":"10.1016\/j.neunet.2026.109208_bib0009","unstructured":"Brantner, B., & Kraus, M. (2023). Symplectic autoencoders for model reduction of Hamiltonian systems. e-print arXiv, arXiv: 2312.10004."},{"issue":"2","key":"10.1016\/j.neunet.2026.109208_bib0010","doi-asserted-by":"crossref","first-page":"A289","DOI":"10.1137\/21M1466657","article-title":"Symplectic model reduction of Hamiltonian systems on nonlinear manifolds and approximation with weakly symplectic autoencoder","volume":"45","author":"Buchfink","year":"2023","journal-title":"SIAM Journal on Scientific Computing"},{"key":"10.1016\/j.neunet.2026.109208_sbref0011","series-title":"Proceedings of the conference algoritmy","first-page":"151","article-title":"PSD-greedy basis generation for structure-preserving model order reduction of Hamiltonian systems","author":"Buchfink","year":"2020"},{"key":"10.1016\/j.neunet.2026.109208_bib0012","doi-asserted-by":"crossref","unstructured":"Cho, K., Van Merri\u00ebnboer, B., Gulcehre, C., Bahdanau, D., Bougares, F., Schwenk, H., & Bengio, Y. (2014). Learning phrase representations using RNN encoder-decoder for statistical machine translation. 10.48550\/arXiv.1406.1078.","DOI":"10.3115\/v1\/D14-1179"},{"key":"10.1016\/j.neunet.2026.109208_bib0013","first-page":"2493","article-title":"Natural language processing (Almost) from scratch","volume":"12","author":"Collobert","year":"2011","journal-title":"Journal of Machine Learning Research"},{"key":"10.1016\/j.neunet.2026.109208_bib0014","series-title":"Lectures on symplectic geometry","author":"da Silva","year":"2008"},{"key":"10.1016\/j.neunet.2026.109208_bib0015","unstructured":"Falcon, W., & The PyTorch Lightning team(2019). PyTorch lightning. https:\/\/github.com\/Lightning-AI\/lightning, 10.5281\/zenodo.3828935."},{"key":"10.1016\/j.neunet.2026.109208_bib0016","series-title":"2020 59th IEEE conference on decision and control (CDC)","first-page":"1190","article-title":"Bayesian identification of Hamiltonian dynamics from symplectic data","author":"Galioto","year":"2020"},{"key":"10.1016\/j.neunet.2026.109208_bib0017","series-title":"Towards understanding the invertibility of convolutional neural networks","author":"Gilbert","year":"2017"},{"issue":"1","key":"10.1016\/j.neunet.2026.109208_bib0018","article-title":"Deep learning for structure-preserving universal stable Koopman-inspired embeddings for nonlinear canonical Hamiltonian dynamics","volume":"6","author":"Goyal","year":"2025","journal-title":"Machine Learning: Science and Technology"},{"key":"10.1016\/j.neunet.2026.109208_sbref0018","series-title":"Proceedings of the 33rd international conference on neural information processing systems","first-page":"15379","article-title":"Hamiltonian neural networks","volume":"vol. 32","author":"Greydanus","year":"2019"},{"key":"10.1016\/j.neunet.2026.109208_bib0020","series-title":"Geometric numerical integration","author":"Hairer","year":"2006"},{"key":"10.1016\/j.neunet.2026.109208_bib0021","series-title":"Deep speech: Scaling up end-to-end speech recognition","author":"Hannun","year":"2014"},{"issue":"6","key":"10.1016\/j.neunet.2026.109208_bib0022","doi-asserted-by":"crossref","first-page":"82","DOI":"10.1109\/MSP.2012.2205597","article-title":"Deep neural networks for acoustic modeling in speech recognition: The shared views of four research groups","volume":"29","author":"Hinton","year":"2012","journal-title":"IEEE Signal Processing Magazine"},{"issue":"5786","key":"10.1016\/j.neunet.2026.109208_bib0023","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1126\/science.1127647","article-title":"Reducing the dimensionality of data with neural networks","volume":"313","author":"Hinton","year":"2006","journal-title":"Science"},{"issue":"4","key":"10.1016\/j.neunet.2026.109208_bib0024","doi-asserted-by":"crossref","first-page":"402","DOI":"10.1007\/s41019-022-00193-5","article-title":"Dimensionality reduction in surrogate modeling: A review of combined methods","volume":"7","author":"Hou","year":"2022","journal-title":"Data Science and Engineering"},{"key":"10.1016\/j.neunet.2026.109208_bib0025","series-title":"Time-adaptive SympNets for separable Hamiltonian systems","author":"Janik","year":"2025"},{"key":"10.1016\/j.neunet.2026.109208_bib0026","doi-asserted-by":"crossref","first-page":"236","DOI":"10.1016\/j.laa.2022.06.009","article-title":"Optimal unit triangular factorization of symplectic matrices","volume":"650","author":"Jin","year":"2022","journal-title":"Linear Algebra and its Applications"},{"key":"10.1016\/j.neunet.2026.109208_bib0027","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1016\/j.neunet.2020.08.017","article-title":"SympNets: Intrinsic structure-preserving symplectic networks for identifying Hamiltonian systems","volume":"132","author":"Jin","year":"2020","journal-title":"Neural Networks"},{"key":"10.1016\/j.neunet.2026.109208_bib0028","unstructured":"Kingma, D. P., & Ba, J. (2017). Adam: A method for stochastic optimization. 10.48550\/arXiv.1412.6980."},{"issue":"6","key":"10.1016\/j.neunet.2026.109208_sbref0028","doi-asserted-by":"crossref","first-page":"84","DOI":"10.1145\/3065386","article-title":"ImageNet classification with deep convolutional neural networks","volume":"60","author":"Krizhevsky","year":"2017","journal-title":"Communications of the ACM"},{"issue":"4","key":"10.1016\/j.neunet.2026.109208_bib0030","doi-asserted-by":"crossref","first-page":"541","DOI":"10.1162\/neco.1989.1.4.541","article-title":"Backpropagation applied to handwritten zip code recognition","volume":"1","author":"LeCun","year":"1989","journal-title":"Neural Computation"},{"key":"10.1016\/j.neunet.2026.109208_bib0031","series-title":"Introduction to mechanics and symmetry: A basic exposition of classical mechanical systems","volume":"vol. 17","author":"Marsden","year":"2013"},{"key":"10.1016\/j.neunet.2026.109208_sbref0031","article-title":"Distributed representations of words and phrases and their compositionality","volume":"26","author":"Mikolov","year":"2013","journal-title":"Advances in Neural Information Processing Systems"},{"key":"10.1016\/j.neunet.2026.109208_bib0033","unstructured":"Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G., Killeen, T., Lin, Z., Gimelshein, N., Antiga, L., Desmaison, A., K\u00f6pf, A., Yang, E., DeVito, Z., Raison, M., Tejani, A., Chilamkurthy, S., Steiner, B., Fang, L., Bai, J., & Chintala, S. (2019). PyTorch: An imperative style, high-performance deep learning library. 10.48550\/arXiv.1912.01703."},{"issue":"1","key":"10.1016\/j.neunet.2026.109208_bib0034","doi-asserted-by":"crossref","first-page":"A1","DOI":"10.1137\/140978922","article-title":"Symplectic model reduction of Hamiltonian systems","volume":"38","author":"Peng","year":"2016","journal-title":"SIAM Journal on Scientific Computing"},{"key":"10.1016\/j.neunet.2026.109208_bib0035","series-title":"Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy","author":"Sharma","year":"2025"},{"key":"10.1016\/j.neunet.2026.109208_bib0036","doi-asserted-by":"crossref","DOI":"10.1016\/j.physd.2021.133122","article-title":"Hamiltonian operator inference: Physics-preserving learning of reduced-order models for canonical Hamiltonian systems","volume":"431","author":"Sharma","year":"2022","journal-title":"Physica D: Nonlinear Phenomena"},{"key":"10.1016\/j.neunet.2026.109208_bib0037","unstructured":"Simonyan, K., & Zisserman, A. (2014). Very deep convolutional networks for large-scale image recognition. 10.48550\/arXiv.1409.1556."},{"key":"10.1016\/j.neunet.2026.109208_bib0038","series-title":"Proceedings of the IEEE conference on computer vision and pattern recognition","first-page":"1","article-title":"Going deeper with convolutions","author":"Szegedy","year":"2015"},{"issue":"2","key":"10.1016\/j.neunet.2026.109208_bib0039","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1016\/0024-3795(72)90024-9","article-title":"The role of symmetric matrices in the study of general matrices","volume":"5","author":"Taussky","year":"1972","journal-title":"Linear Algebra and its Applications"},{"issue":"1\u20133","key":"10.1016\/j.neunet.2026.109208_bib0040","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1016\/0169-7439(87)80084-9","article-title":"Principal component analysis","volume":"2","author":"Wold","year":"1987","journal-title":"Chemometrics and Intelligent Laboratory Systems"},{"issue":"2","key":"10.1016\/j.neunet.2026.109208_bib0041","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1615\/JMachLearnModelComput.2024052810","article-title":"Data-driven identification of quadratic representations for nonlinear Hamiltonian systems using weakly symplectic liftings","volume":"5","author":"Y\u0131ld\u0131z","year":"2024","journal-title":"Journal of Machine Learning for Modeling and Computing"},{"key":"10.1016\/j.neunet.2026.109208_bib0042","doi-asserted-by":"crossref","unstructured":"Y\u0131ld\u0131z, S., Janik, K., & Benner, P. (2025). Symplectic convolutional neural networks. Zenodo. 10.5281\/zenodo.16962444.","DOI":"10.1016\/j.neunet.2026.109208"}],"container-title":["Neural Networks"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0893608026006696?httpAccept=text\/xml","content-type":"text\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/api.elsevier.com\/content\/article\/PII:S0893608026006696?httpAccept=text\/plain","content-type":"text\/plain","content-version":"vor","intended-application":"text-mining"}],"deposited":{"date-parts":[[2026,6,7]],"date-time":"2026-06-07T19:59:00Z","timestamp":1780862340000},"score":1,"resource":{"primary":{"URL":"https:\/\/linkinghub.elsevier.com\/retrieve\/pii\/S0893608026006696"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,11]]},"references-count":42,"alternative-id":["S0893608026006696"],"URL":"https:\/\/doi.org\/10.1016\/j.neunet.2026.109208","relation":{},"ISSN":["0893-6080"],"issn-type":[{"value":"0893-6080","type":"print"}],"subject":[],"published":{"date-parts":[[2026,11]]},"assertion":[{"value":"Elsevier","name":"publisher","label":"This article is maintained by"},{"value":"Symplectic convolutional neural networks","name":"articletitle","label":"Article Title"},{"value":"Neural Networks","name":"journaltitle","label":"Journal Title"},{"value":"https:\/\/doi.org\/10.1016\/j.neunet.2026.109208","name":"articlelink","label":"CrossRef DOI link to publisher maintained version"},{"value":"article","name":"content_type","label":"Content Type"},{"value":"\u00a9 2026 The Author(s). Published by Elsevier Ltd.","name":"copyright","label":"Copyright"}],"article-number":"109208"}}