Learning Coarse-Graining Transformations in the 2D Ising Model: A Physics-Informed Approach to Neural Network Interpretability

Learning Coarse-Graining Transformations in the 2D Ising Model: A Physics-Informed Approach to Neural Network Interpretability | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Learning Coarse-Graining Transformations in the 2D Ising Model: A Physics-Informed Approach to Neural Network Interpretability Vansh Agrawal, Mayank Goswami This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8202104/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Understanding how neural networks learn to abstract and simplify information is a fundamental challenge in AI interpretability. We demonstrate that simple physics models can provide a rigorous testbed for evaluating learned representations. Using the 2D Ising model as a case study, we train a convolutional neu ral network to perform coarse-graining—a fundamental operation in statistical physics that reduces degrees of freedom while preserving essential macroscopic proper ties. Our CNN achieves near-perfect performance (MSE = 0.020, spatial correlation = 0.9997) with only 9,569 parameters, outperforming a fully-connected MLP by 648× despite using 32× fewer parameters. This work es tablishes physics-based coarse-graining as a benchmark for evaluating inherent biases in neural network archi tectures, revealing how spatial inductive biases enable efficient learning of physical abstractions. Artificial Intelligence and Machine Learning Mathematical Physics Physics informed ANN Full Text Additional Declarations The authors declare no competing interests. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-8202104","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":550499983,"identity":"42c48a67-3db9-4258-8b68-c1fdc4440e84","order_by":0,"name":"Vansh Agrawal","email":"","orcid":"","institution":"Indian institute of technology Roorkee","correspondingAuthor":false,"prefix":"","firstName":"Vansh","middleName":"","lastName":"Agrawal","suffix":""},{"id":550499984,"identity":"1ec26cae-6869-4020-8f86-1212c5285f7a","order_by":1,"name":"Mayank Goswami","email":"data:image/png;base64,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","orcid":"https://orcid.org/0000-0001-8829-9980","institution":"Indian institute of technology Roorkee","correspondingAuthor":true,"prefix":"","firstName":"Mayank","middleName":"","lastName":"Goswami","suffix":""}],"badges":[],"createdAt":"2025-11-25 10:32:46","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":false,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":false},"doi":"10.21203/rs.3.rs-8202104/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8202104/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":96916203,"identity":"691d7a0a-2c49-4442-b9e6-2987e5368edb","added_by":"auto","created_at":"2025-11-27 14:08:12","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":857003,"visible":true,"origin":"","legend":"","description":"","filename":"LearningCoarseGrainingTransformationsinthe2DIsingModelA.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8202104/v1_covered_9abde5c1-9b3a-4340-9e4e-0f782d9ad1ee.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eLearning Coarse-Graining Transformations in the 2D Ising Model: A Physics-Informed Approach to Neural Network Interpretability\u003c/p\u003e","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"Indian Institute of Technology Roorkee","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Physics informed ANN","lastPublishedDoi":"10.21203/rs.3.rs-8202104/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8202104/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eUnderstanding how neural networks learn \u0026nbsp;to abstract and simplify information is a fundamental \u0026nbsp;challenge in AI interpretability. We demonstrate that \u0026nbsp;simple physics models can provide a rigorous testbed for \u0026nbsp;evaluating learned representations. Using the 2D Ising \u0026nbsp;model as a case study, we train a convolutional neu ral network to perform coarse-graining—a fundamental \u0026nbsp;operation in statistical physics that reduces degrees of \u0026nbsp;freedom while preserving essential macroscopic proper ties. Our CNN achieves near-perfect performance (MSE \u0026nbsp;= 0.020, spatial correlation = 0.9997) with only 9,569 \u0026nbsp;parameters, outperforming a fully-connected MLP by \u0026nbsp;648× despite using 32× fewer parameters. This work es tablishes physics-based coarse-graining as a benchmark \u0026nbsp;for evaluating inherent biases in neural network archi tectures, revealing how spatial inductive biases enable \u0026nbsp;efficient learning of physical abstractions.\u003c/p\u003e","manuscriptTitle":"Learning Coarse-Graining Transformations in the 2D Ising Model: A Physics-Informed Approach to Neural Network Interpretability","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-11-26 02:59:11","doi":"10.21203/rs.3.rs-8202104/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"32bb5b39-03bc-424a-b677-0dc52ecfada0","owner":[],"postedDate":"November 26th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[{"id":58573183,"name":"Artificial Intelligence and Machine Learning"},{"id":58573184,"name":"Mathematical Physics"}],"tags":[],"updatedAt":"2025-11-26T02:59:11+00:00","versionOfRecord":[],"versionCreatedAt":"2025-11-26 02:59:11","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8202104","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8202104","identity":"rs-8202104","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}