{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T12:05:57Z","timestamp":1779192357097,"version":"3.51.4"},"reference-count":64,"publisher":"AIP Publishing","issue":"19","funder":[{"DOI":"10.13039\/100007569","name":"Carl-Zeiss-Stiftung","doi-asserted-by":"publisher","id":[{"id":"10.13039\/100007569","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["233630050"],"award-info":[{"award-number":["233630050"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["465145163"],"award-info":[{"award-number":["465145163"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["405553726"],"award-info":[{"award-number":["405553726"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["278162697"],"award-info":[{"award-number":["278162697"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["pubs.aip.org"],"crossmark-restriction":true},"short-container-title":[],"published-print":{"date-parts":[[2026,5,21]]},"abstract":"<jats:p>Localized topological defects inherently possess a multiscale character. While their microstructure configuration depends on the specific physical system, their topological features and mutual interactions can be described on the macroscale in terms of a particle representation. However, determining the physical properties associated with a given defect pattern often requires knowledge of the underlying microscopic structure. In this study, we extend a Wasserstein generative adversarial neural network by incorporating physical constraints and Fourier-space information to generate microscopic spin configurations consistent with prescribed macroscopic patterns and thermodynamic parameters. Using the two-dimensional XY model as a test case, where vortex\u2013antivortex pairs act as long-range interacting defects, we show that the model generates spin configurations that accurately reproduce magnetization, susceptibility, helicity modulus, and spin\u2013spin correlations over a wide range of temperatures below the Kosterlitz\u2013Thouless transition. At the same time, deviations in the specific heat reveal limitations in reproducing higher-order energy fluctuations. A complementary analysis based on topological data analysis uncovers subtle differences in global spin-correlation structures at near-critical temperatures that are not apparent from conventional correlation functions alone. These results demonstrate both the promise and current limitations of generative approaches for multiscale studies of defect-dominated spin systems and, at the same time, highlight topological methods as valuable tools for characterizing critical behavior.<\/jats:p>","DOI":"10.1063\/5.0323442","type":"journal-article","created":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T11:13:03Z","timestamp":1779189183000},"update-policy":"https:\/\/doi.org\/10.1063\/aip-crossmark-policy-page","source":"Crossref","is-referenced-by-count":0,"title":["Reconstruction of spin structures from topological charge distributions via generative neural network systems"],"prefix":"10.1063","volume":"164","author":[{"ORCID":"https:\/\/orcid.org\/0009-0002-1895-8439","authenticated-orcid":false,"given":"Kyra H. M.","family":"Klos","sequence":"first","affiliation":[{"name":"Institute of Physics, Johannes Gutenberg-University Mainz 1 , 55128 Mainz,","place":["Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0379-5673","authenticated-orcid":false,"given":"Jan","family":"Disselhoff","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, Johannes Gutenberg-University Mainz 2 , 55128 Mainz,","place":["Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0004-0509-789X","authenticated-orcid":false,"given":"Michael","family":"Wand","sequence":"additional","affiliation":[{"name":"Institute of Computer Science, Johannes Gutenberg-University Mainz 2 , 55128 Mainz,","place":["Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8767-6633","authenticated-orcid":false,"given":"Karin","family":"Everschor-Sitte","sequence":"additional","affiliation":[{"name":"Faculty of Physics and Center for Nanointegration Duisburg-Essen (CENIDE), University of Duisburg-Essen 3 , 47057 Duisburg,","place":["Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5536-6718","authenticated-orcid":false,"given":"Friederike","family":"Schmid","sequence":"additional","affiliation":[{"name":"Institute of Physics, Johannes Gutenberg-University Mainz 1 , 55128 Mainz,","place":["Germany"]}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"317","published-online":{"date-parts":[[2026,5,19]]},"reference":[{"key":"2026051907125900300_c1","doi-asserted-by":"publisher","first-page":"209","DOI":"10.1146\/annurev.ns.43.120193.001233","article-title":"Effective field theory","volume":"43","year":"1993","journal-title":"Annu. Rev. Nucl. Part. Sci."},{"issue":"2","key":"2026051907125900300_c2","doi-asserted-by":"publisher","first-page":"879","DOI":"10.1021\/nn102099z","article-title":"Exploring topological defects in epitaxial BiFeO3 thin films","volume":"5","year":"2011","journal-title":"ACS Nano"},{"issue":"5916","key":"2026051907125900300_c3","doi-asserted-by":"publisher","first-page":"915","DOI":"10.1126\/science.1166767","article-title":"Skyrmion lattice in a chiral magnet","volume":"323","year":"2009","journal-title":"Science"},{"issue":"25\u201326","key":"2026051907125900300_c4","doi-asserted-by":"publisher","first-page":"4117","DOI":"10.1080\/14786430600593016","article-title":"Topological point defects in nematic liquid crystals","volume":"86","year":"2006","journal-title":"Philos. Mag."},{"key":"2026051907125900300_c5","doi-asserted-by":"publisher","first-page":"9280","DOI":"10.1039\/c6sm01748g","article-title":"Topological defects in cholesteric liquid crystal shells","volume":"12","year":"2016","journal-title":"Soft Matter"},{"issue":"6314","key":"2026051907125900300_c6","doi-asserted-by":"publisher","first-page":"882","DOI":"10.1126\/science.aah6936","article-title":"Command of active matter by topological defects and patterns","volume":"354","year":"2016","journal-title":"Science"},{"key":"2026051907125900300_c7","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1038\/s42254-022-00445-3","article-title":"Topological active matter","volume":"4","year":"2022","journal-title":"Nat. Rev. Phys."},{"key":"2026051907125900300_c8","doi-asserted-by":"publisher","first-page":"260501","DOI":"10.1063\/5.0148469","article-title":"Perspective on unconventional computing using magnetic skyrmions","volume":"122","year":"2023","journal-title":"Appl. Phys. Lett."},{"key":"2026051907125900300_c9","doi-asserted-by":"publisher","first-page":"040501","DOI":"10.1103\/RevModPhys.89.040501","article-title":"Topological defects and phase transitions","volume":"89","year":"2017","journal-title":"Rev. Mod. Phys."},{"key":"2026051907125900300_c10","doi-asserted-by":"publisher","first-page":"112","DOI":"10.3389\/fphy.2020.00112","article-title":"Topological point defects of liquid crystals in quasi-two-dimensional geometries","volume":"8","year":"2020","journal-title":"Front. Phys."},{"key":"2026051907125900300_c11","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1039\/c8sm01901k","article-title":"Theory of defect motion in 2D passive and active nematic liquid crystals","volume":"15","year":"2019","journal-title":"Soft Matter"},{"key":"2026051907125900300_c12","doi-asserted-by":"publisher","first-page":"046701","DOI":"10.1103\/physrevlett.134.046701","article-title":"Realizing quantitative quasiparticle modeling of skyrmion dynamics in arbitrary potentials","volume":"134","year":"2025","journal-title":"Phys. Rev. Lett."},{"key":"2026051907125900300_c13","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1103\/physrevlett.30.230","article-title":"Steady-state motion of magnetic domains","volume":"30","year":"1973","journal-title":"Phys. Rev. Lett."},{"key":"2026051907125900300_c14","doi-asserted-by":"publisher","first-page":"1751","DOI":"10.1039\/c9sm01979k","article-title":"End-to-end machine learning for experimental physics: Using simulated data to train a neural network for object detection in video microscopy","volume":"16","year":"2020","journal-title":"Soft Matter"},{"key":"2026051907125900300_c15","doi-asserted-by":"publisher","first-page":"013259","DOI":"10.1103\/physrevresearch.6.013259","article-title":"Machine eye for defects: Machine learning-based solution to identify and characterize topological defects in textured images of nematic materials","volume":"6","year":"2024","journal-title":"Phys. Rev. Res."},{"key":"2026051907125900300_c16","doi-asserted-by":"publisher","first-page":"045207","DOI":"10.1103\/physrevb.97.045207","article-title":"Machine learning vortices at the Kosterlitz-Thouless transition","volume":"97","year":"2018","journal-title":"Phys. Rev. B"},{"key":"2026051907125900300_c17","doi-asserted-by":"publisher","first-page":"062701","DOI":"10.1103\/physreve.99.062701","article-title":"Machine learning topological defects of confined liquid crystals in two dimensions","volume":"99","year":"2019","journal-title":"Phys. Rev. E"},{"key":"2026051907125900300_c18","doi-asserted-by":"publisher","first-page":"1869","DOI":"10.1039\/d3sm01253k","article-title":"A machine learning approach to robustly determine director fields and analyze defects in active nematics","volume":"20","year":"2024","journal-title":"Soft Matter"},{"key":"2026051907125900300_c19","doi-asserted-by":"publisher","first-page":"102171","DOI":"10.1016\/j.isci.2021.102171","article-title":"A deeper look into natural sciences with physics-based and data-driven measures","volume":"24","year":"2021","journal-title":"iScience"},{"issue":"2","key":"2026051907125900300_c20","doi-asserted-by":"publisher","first-page":"267","DOI":"10.2140\/camcos.2021.16.267","article-title":"Scalable computational measures for entropic detection of latent relations and their applications to magnetic imaging","volume":"16","year":"2021","journal-title":"Commun. Appl. Math. Comput. Sci."},{"key":"2026051907125900300_c21","unstructured":"M.\n            Cristoforetti\n          , G.Jurman, A. I.Nardelli, and C.Furlanello, \u201cTowards meaningful physics from generative models,\u201d arXiv:1705.09524 (2017)."},{"key":"2026051907125900300_c22","unstructured":"A.\n            Naravane\n           and N.Mathur, \u201cSemi-supervised learning of order parameter in 2D Ising and XY models using conditional variational autoencoders,\u201d arXiv:2306.16822 (2023)."},{"key":"2026051907125900300_c23","doi-asserted-by":"publisher","first-page":"024112","DOI":"10.1103\/physrevb.111.024112","article-title":"Learning thermodynamics and topological order of the two-dimensional XY model with generative real-valued restricted Boltzmann machines","volume":"111","year":"2025","journal-title":"Phys. Rev. B"},{"issue":"1","key":"2026051907125900300_c24","doi-asserted-by":"publisher","first-page":"20377","DOI":"10.1038\/s41598-023-47866-3","article-title":"Topological magnetic structure generation using VAE-GAN hybrid model and discriminator-driven latent sampling","volume":"13","year":"2023","journal-title":"Sci. Rep."},{"key":"2026051907125900300_c25","doi-asserted-by":"publisher","first-page":"1181","DOI":"10.1088\/0022-3719\/6\/7\/010","article-title":"Ordering, metastability and phase transitions in two-dimensional systems","volume":"6","year":"1973","journal-title":"J.\u00a0Phys. C: Solid State Phys."},{"key":"2026051907125900300_c26","first-page":"493","article-title":"Destruction of long-range order in one-dimensional and two-dimensional systems having a continuous symmetry group I. Classical systems","volume":"32","year":"1970","journal-title":"J.\u00a0Exp. Theor. Phys."},{"issue":"2","key":"2026051907125900300_c27","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1016\/0370-1573(79)90116-9","article-title":"XVI. Superfluidity and the two dimensional XY model","volume":"49","year":"1979","journal-title":"Phys. Rep."},{"key":"2026051907125900300_c28","volume-title":"Principles of Condensed Matter Physics","year":"1995"},{"key":"2026051907125900300_c29","doi-asserted-by":"publisher","first-page":"1133","DOI":"10.1103\/physrevlett.17.1133","article-title":"Absence of ferromagnetism or antiferromagnetism in one- or two-dimensional isotropic heisenberg models","volume":"17","year":"1966","journal-title":"Phys. Rev. Lett."},{"key":"2026051907125900300_c30","first-page":"19","article-title":"On the theory of phase transitions","volume":"7","year":"1937","journal-title":"J.\u00a0Exp. Theor. Phys."},{"issue":"26","key":"2026051907125900300_c31","doi-asserted-by":"publisher","first-page":"5869","DOI":"10.1088\/0305-4470\/38\/26\/003","article-title":"The two-dimensional XY model at the Berezinskii-Kosterlitz-Thouless transition","volume":"38","year":"2005","journal-title":"J.\u00a0Phys. A: Math. Gen."},{"key":"2026051907125900300_c32","doi-asserted-by":"publisher","first-page":"1111","DOI":"10.1103\/physreva.8.1111","article-title":"Helicity modulus, superfluidity, and scaling in isotropic systems","volume":"8","year":"1973","journal-title":"Phys. Rev. A"},{"key":"2026051907125900300_c33","doi-asserted-by":"publisher","first-page":"693","DOI":"10.1103\/physrevlett.47.693","article-title":"Critical properties from Monte Carlo coarse graining and renormalization","volume":"47","year":"1981","journal-title":"Phys. Rev. Lett."},{"key":"2026051907125900300_c34","first-page":"505","volume-title":"Geometry, Topology and Physics","year":"1990; 2003"},{"key":"2026051907125900300_c35","article-title":"Persistent homology\u2014a survey","volume-title":"Discrete & Computational Geometry - DCG","year":"2008"},{"key":"2026051907125900300_c36","doi-asserted-by":"publisher","first-page":"024121","DOI":"10.1103\/physreve.105.024121","article-title":"Quantitative analysis of phase transitions in two-dimensional XY models using persistent homology","volume":"105","year":"2022","journal-title":"Phys. Rev. E"},{"key":"2026051907125900300_c37","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1007\/s00454-002-2885-2","article-title":"Topological persistence and simplification","volume":"28","year":"2002","journal-title":"Discrete Comput. Geom."},{"key":"2026051907125900300_c38","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1140\/epjds\/s13688-017-0109-5","article-title":"A roadmap for the computation of persistent homology","volume":"6","year":"2017","journal-title":"EPJ Data Sci."},{"issue":"8","key":"2026051907125900300_c39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.5555\/3122009.3122017","article-title":"Persistence images: A stable vector representation of persistent homology","volume":"18","year":"2017","journal-title":"J.\u00a0Mach. Learn. Res."},{"key":"2026051907125900300_c40","article-title":"Scikit-tda: Topological data analysis for python","year":"2019"},{"key":"2026051907125900300_c41","doi-asserted-by":"publisher","first-page":"697","DOI":"10.48550\/arXiv.astro-ph\/9312028","article-title":"Robust morphological measures for large scale structure in the universe","volume":"288","year":"1994","journal-title":"Astron. Astrophys."},{"key":"2026051907125900300_c42","doi-asserted-by":"publisher","first-page":"1583","DOI":"10.5555\/3495724.3495858","article-title":"Neural networks fail to learn periodic functions and how to fix it","volume":"33","year":"2020","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"2026051907125900300_c43","first-page":"8024","article-title":"Pytorch: An imperative style, high-performance deep learning library","volume":"32","year":"2019","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"2026051907125900300_c44","first-page":"214","article-title":"Wasserstein generative adversarial networks","year":"2017"},{"key":"2026051907125900300_c45","article-title":"Generative adversarial nets","volume-title":"Advances in Neural Information Processing Systems","author":"Ghahramani","year":"2014"},{"key":"2026051907125900300_c46","first-page":"1","volume-title":"Mode collapse in generative adversarial networks: An overview","year":"2022"},{"key":"2026051907125900300_c47","article-title":"Does diffusion beat GAN in image super resolution?","year":"2024"},{"key":"2026051907125900300_c48","first-page":"234","article-title":"U-net: Convolutional networks for biomedical image segmentation","volume-title":"Medical Image Computing and Computer-Assisted Intervention (MICCAI), Vol. 9351 of LNCS","year":"2015"},{"key":"2026051907125900300_c49","first-page":"1510","article-title":"Arbitrary style transfer in real-time with adaptive instance normalization","year":"2017"},{"key":"2026051907125900300_c50","first-page":"4401","article-title":"A style-based generator architecture for generative adversarial networks","year":"2019"},{"key":"2026051907125900300_c51","first-page":"807","article-title":"Rectified linear units improve restricted Boltzmann machines,\u201d in International Conference on Machine Learning (Omni","year":"2010"},{"key":"2026051907125900300_c52","first-page":"3","article-title":"Rectifier nonlinearities improve neural network acoustic models","year":"2013"},{"key":"2026051907125900300_c53","volume-title":"Deconvolution and Checkerboard Artifacts","year":"2016"},{"key":"2026051907125900300_c54","doi-asserted-by":"publisher","first-page":"2672","DOI":"10.1145\/3422622","article-title":"Generative adversarial networks","volume":"27","year":"2014","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"2026051907125900300_c55","first-page":"52","article-title":"On a space of completely additive functions","volume":"13","year":"1958","journal-title":"Vestn. Leningr. Univ."},{"key":"2026051907125900300_c56","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"2026051907125900300_c57","volume-title":"Topics in Optimal Transportation","year":"2003"},{"key":"2026051907125900300_c58","year":"2017"},{"key":"2026051907125900300_c59","article-title":"Evaluation of mode collapse in generative adversarial networks","year":"2018"},{"key":"2026051907125900300_c60","first-page":"4278","article-title":"Inception-v4, inception-resnet and the impact of residual connections on learning","year":"2017"},{"key":"2026051907125900300_c61","first-page":"770","volume-title":"Deep residual learning for image recognition","year":"2016"},{"key":"2026051907125900300_c63","article-title":"Adam: A method for stochastic optimization","year":"2014"},{"key":"2026051907125900300_c64","first-page":"7887","article-title":"Watch your up-convolution: CNN based generative deep neural networks are failing to reproduce spectral distributions","year":"2020"},{"key":"2026051907125900300_c65","first-page":"2020","article-title":"Spectral distribution aware image generation","volume-title":"AAAI Conference on Artificial Intelligence"}],"container-title":["The Journal of Chemical Physics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0323442\/21009192\/194113_1_5.0323442.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"syndication"},{"URL":"https:\/\/pubs.aip.org\/aip\/jcp\/article-pdf\/doi\/10.1063\/5.0323442\/21009192\/194113_1_5.0323442.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,19]],"date-time":"2026-05-19T11:13:08Z","timestamp":1779189188000},"score":1,"resource":{"primary":{"URL":"https:\/\/pubs.aip.org\/jcp\/article\/164\/19\/194113\/3391587\/Reconstruction-of-spin-structures-from-topological"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,5,19]]},"references-count":64,"journal-issue":{"issue":"19","published-print":{"date-parts":[[2026,5,21]]}},"URL":"https:\/\/doi.org\/10.1063\/5.0323442","relation":{},"ISSN":["0021-9606","1089-7690"],"issn-type":[{"value":"0021-9606","type":"print"},{"value":"1089-7690","type":"electronic"}],"subject":[],"published-other":{"date-parts":[[2026,5,21]]},"published":{"date-parts":[[2026,5,19]]},"article-number":"194113"}}