{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,15]],"date-time":"2026-06-15T14:21:44Z","timestamp":1781533304387,"version":"3.54.5"},"reference-count":40,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T00:00:00Z","timestamp":1774915200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Key Research and Development Program of China","award":["2024YFA1611003"],"award-info":[{"award-number":["2024YFA1611003"]}]},{"award":["2024YFA1611003"],"award-info":[{"award-number":["2024YFA1611003"]}],"id":[{"id":"https:\/\/ror.org\/024zcda42","id-type":"ROR","asserted-by":"publisher"}]},{"name":"the Fundamental Research Funds for Central China Normal University","award":["CCNU24JC007"],"award-info":[{"award-number":["CCNU24JC007"]}]},{"award":["CCNU24JC007"],"award-info":[{"award-number":["CCNU24JC007"]}],"id":[{"id":"https:\/\/ror.org\/03x1jna21","id-type":"ROR","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012137","name":"Key Laboratory of Quark and Lepton Physics","doi-asserted-by":"crossref","award":["QLPL2025P01"],"award-info":[{"award-number":["QLPL2025P01"]}],"id":[{"id":"10.13039\/501100012137","id-type":"DOI","asserted-by":"crossref"}]},{"name":"the 111 Project","award":["BP0820038"],"award-info":[{"award-number":["BP0820038"]}]},{"DOI":"10.13039\/501100012226","name":"Fundamental Research Funds for the Central Universities","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100012226","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The geometric signatures of macroscopic interfaces in the two-dimensional critical Ising model strictly adhere to Schramm\u2013Loewner Evolution (SLE) theory. In this study, we propose a physics-driven generative approach using Super-Resolution Generative Adversarial Networks (SRGANs) to approximate the inverse coarse-graining operation to generate larger configurations. From the perspective of Geometric Deep Learning (GDL), we leverage the geometric priors of Convolutional Neural Networks (CNNs)\u2014specifically their translational and rotational symmetries\u2014to effectively encode the universal physical laws of the Ising Hamiltonian. This inductive bias allows the model to be trained on small scales yet be generalized to large-scale systems (2048\u00a0\u00d7\u00a02048) while preserving physical conservation. To accommodate spin discreteness, we employ an L1-based loss function to maintain domain wall sharpness. SLE analysis and long-range correlation functions confirm that the model reproduces critical dynamics and conformal invariance, successfully serving as a physics-preserving inverse coarse-graining transformation framework.<\/jats:p>","DOI":"10.3390\/e28040385","type":"journal-article","created":{"date-parts":[[2026,3,31]],"date-time":"2026-03-31T16:34:08Z","timestamp":1774974848000},"page":"385","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Generating the Critical Ising Model via SRGAN: A Schramm\u2013Loewner Evolution Analysis from a Geometric Deep Learning Perspective"],"prefix":"10.3390","volume":"28","author":[{"ORCID":"https:\/\/orcid.org\/0009-0000-9069-0591","authenticated-orcid":false,"given":"Yuxiang","family":"Yang","sequence":"first","affiliation":[{"name":"Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Wei","family":"Li","sequence":"additional","affiliation":[{"name":"Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China"},{"name":"SCIQ Lab, ESIEA, Campus Ivry sur Seine, 73 bis Avenue Maurice Thorez, 94200 Ivry sur Seine, France"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0003-9852-176X","authenticated-orcid":false,"given":"Yanyang","family":"Wang","sequence":"additional","affiliation":[{"name":"Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhihang","family":"Liu","sequence":"additional","affiliation":[{"name":"Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China"},{"name":"Wuhan Technology and Business University, Wuhan 430065, China"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kui","family":"Tuo","sequence":"additional","affiliation":[{"name":"Key Laboratory of Quark and Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,31]]},"reference":[{"key":"ref_1","unstructured":"Henkel, M. (2013). Conformal Invariance and Critical Phenomena, Springer Science & Business Media."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"L201","DOI":"10.1088\/0305-4470\/25\/4\/009","article-title":"Critical percolation in finite geometries","volume":"25","author":"Cardy","year":"1992","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1070\/SM1967v002n03ABEH002345","article-title":"The phenomenon of \u201cphase separation\u201d at low temperatures in some lattice models of a gas. I","volume":"2","author":"Minlos","year":"1967","journal-title":"Math. USSR-Sb."},{"key":"ref_4","first-page":"113","article-title":"The phenomenon of \u201cseparation of phases\u201d at low temperatures in certain lattice models of a gas. II","volume":"19","author":"Minlos","year":"1968","journal-title":"Tr. Mosk. Mat. Obs."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/BF01645615","article-title":"The phase separation line in the two-dimensional Ising model","volume":"27","author":"Gallavotti","year":"1972","journal-title":"Commun. Math. Phys."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"313","DOI":"10.1007\/BF01013174","article-title":"On the local structure of the phase separation line in the two-dimensional Ising system","volume":"26","author":"Bricmont","year":"1981","journal-title":"J. Stat. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1007\/BF02803524","article-title":"Scaling limits of loop-erased random walks and uniform spanning trees","volume":"118","author":"Schramm","year":"2000","journal-title":"Isr. J. Math."},{"key":"ref_8","unstructured":"Rohde, S., and Schramm, O. (2011). Selected Works of Oded Schramm, Springer."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Smirnov, S. (2010). Conformal invariance in random cluster models. I. Holmorphic fermions in the Ising model. Annals of Mathematics, Mathematics Department, Princeton University.","DOI":"10.4007\/annals.2010.172.1435"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"157","DOI":"10.1016\/j.crma.2013.12.002","article-title":"Convergence of Ising interfaces to Schramm\u2019s SLE curves","volume":"352","author":"Chelkak","year":"2014","journal-title":"C. R. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1038\/nphys4035","article-title":"Machine learning phases of matter","volume":"13","author":"Carrasquilla","year":"2017","journal-title":"Nat. Phys."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"195105","DOI":"10.1103\/PhysRevB.94.195105","article-title":"Discovering phase transitions with unsupervised learning","volume":"94","author":"Wang","year":"2016","journal-title":"Phys. Rev. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"052140","DOI":"10.1103\/PhysRevE.103.052140","article-title":"Supervised and unsupervised learning of directed percolation","volume":"103","author":"Shen","year":"2021","journal-title":"Phys. Rev. E"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"032142","DOI":"10.1103\/PhysRevE.99.032142","article-title":"Machine learning of phase transitions in the percolation and XY models","volume":"99","author":"Zhang","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"130179","DOI":"10.1016\/j.physleta.2024.130179","article-title":"Identifying percolation phase transitions with unsupervised learning based on largest clusters","volume":"531","author":"Xu","year":"2025","journal-title":"Phys. Lett. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"024102","DOI":"10.1103\/PhysRevE.110.024102","article-title":"Supervised, semisupervised, and unsupervised learning of the Domany-Kinzel model","volume":"110","author":"Tuo","year":"2024","journal-title":"Phys. Rev. E"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"015051","DOI":"10.1088\/2632-2153\/adb370","article-title":"Autoencoder-assisted study of directed percolation with spatial long-range interactions","volume":"6","author":"Wang","year":"2025","journal-title":"Mach. Learn. Sci. Technol."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"275701","DOI":"10.1103\/PhysRevLett.89.275701","article-title":"Inverse Monte Carlo renormalization group transformations for critical phenomena","volume":"89","author":"Ron","year":"2002","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"075113","DOI":"10.1103\/PhysRevB.99.075113","article-title":"Super-resolving the Ising model with convolutional neural networks","volume":"99","author":"Efthymiou","year":"2019","journal-title":"Phys. Rev. B"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"9617","DOI":"10.1038\/s41598-021-88605-w","article-title":"Inverse renormalization group based on image super-resolution using deep convolutional networks","volume":"11","author":"Shiina","year":"2021","journal-title":"Sci. Rep."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Blau, Y., and Michaeli, T. (2018). The perception-distortion tradeoff. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, IEEE.","DOI":"10.1109\/CVPR.2018.00652"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"81","DOI":"10.1007\/s00138-022-01338-2","article-title":"Ising granularity image analysis on VAE\u2013GAN","volume":"33","author":"Chen","year":"2022","journal-title":"Mach. Vis. Appl."},{"key":"ref_23","unstructured":"Liu, Z., Rodrigues, S.P., and Cai, W. (2017). Simulating the Ising model with a deep convolutional generative adversarial network. arXiv."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Ledig, C., Theis, L., Husz\u00e1r, F., Caballero, J., Cunningham, A., Acosta, A., Aitken, A., Tejani, A., Totz, J., and Wang, Z. (2017). Photo-realistic single image super-resolution using a generative adversarial network. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, Honolulu, HI, USA, 21\u201326 July 2017, IEEE.","DOI":"10.1109\/CVPR.2017.19"},{"key":"ref_25","unstructured":"Cohen, T., and Welling, M. (2016). Group equivariant convolutional networks. International Conference on Machine Learning, PMLR."},{"key":"ref_26","unstructured":"Bronstein, M.M., Bruna, J., Cohen, T., and Velickovic, P. (2021). Geometric deep learning: Grids, groups, graphs, geodesics, and gauges. arXiv."},{"key":"ref_27","unstructured":"Mehta, P., and Schwab, D.J. (2014). An exact mapping between the variational renormalization group and deep learning. arXiv."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1223","DOI":"10.1007\/s10955-017-1836-5","article-title":"Why does deep and cheap learning work so well?","volume":"168","author":"Lin","year":"2017","journal-title":"J. Stat. Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"117","DOI":"10.1103\/PhysRev.65.117","article-title":"Crystal statistics. I. A two-dimensional model with an order-disorder transition","volume":"65","author":"Onsager","year":"1944","journal-title":"Phys. Rev."},{"key":"ref_30","unstructured":"Binder, K. (2022). Statistical and Nonlinear Physics, Springer."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1023\/B:JOSS.0000013973.40984.3b","article-title":"The Loewner equation: Maps and shapes","volume":"114","author":"Gruzberg","year":"2004","journal-title":"J. Stat. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"433","DOI":"10.5802\/afst.1056","article-title":"Discrete L\u00f6wner evolution","volume":"12","author":"Bauer","year":"2003","journal-title":"Ann. Fac. Sci. Toulouse Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"803","DOI":"10.1007\/s10955-008-9535-x","article-title":"Computing the Loewner driving process of random curves in the half plane","volume":"131","author":"Kennedy","year":"2008","journal-title":"J. Stat. Phys."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1115","DOI":"10.1007\/s10955-010-0031-8","article-title":"Exact sampling of self-avoiding paths via discrete Schramm-Loewner evolution","volume":"140","author":"Gherardi","year":"2010","journal-title":"J. Stat. Phys."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"839","DOI":"10.1007\/s10955-009-9866-2","article-title":"Numerical computations for the Schramm-Loewner evolution","volume":"137","author":"Kennedy","year":"2009","journal-title":"J. Stat. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1421","DOI":"10.1214\/07-AOP364","article-title":"The dimension of the SLE curves","volume":"36","author":"Beffara","year":"2008","journal-title":"Ann. Probab."},{"key":"ref_37","unstructured":"Chatelain, C. (2012). Conformal Invariance: An Introduction to Loops, Interfaces and Stochastic Loewner Evolution, Springer."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1016\/j.physrep.2006.06.002","article-title":"2D growth processes: SLE and Loewner chains","volume":"432","author":"Bauer","year":"2006","journal-title":"Phys. Rep."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"P12009","DOI":"10.1088\/1742-5468\/2005\/12\/P12009","article-title":"The scaling limit of two cluster boundaries in critical lattice models","volume":"2005","author":"Gamsa","year":"2005","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"020403","DOI":"10.1103\/PhysRevB.76.020403","article-title":"Possible description of domain walls in two-dimensional spin glasses by stochastic Loewner evolutions","volume":"76","author":"Bernard","year":"2007","journal-title":"Phys. Rev. B\u2014Condens. Matter Mater. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/4\/385\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,3]],"date-time":"2026-04-03T04:34:43Z","timestamp":1775190883000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/28\/4\/385"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,3,31]]},"references-count":40,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2026,4]]}},"alternative-id":["e28040385"],"URL":"https:\/\/doi.org\/10.3390\/e28040385","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,3,31]]}}}