A unified physics-informed generative operator framework for general inverse problems

A unified physics-informed generative operator framework for general inverse problems | 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 Article A unified physics-informed generative operator framework for general inverse problems Yaohua Zang, Gang Bao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7867321/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted You are reading this latest preprint version Abstract Solving inverse problems governed by partial differential equations (PDEs) is central to science and engineering, yet remains challenging when measurements are sparse, noisy, or when the underlying coefficients are high-dimensional or discontinuous. Existing deep learning approaches either require extensive labeled datasets or are limited to specific measurement types, often leading to failure in such regimes and restricting their practical applicability. Here, a novel generative neural operator framework, IGNO, is introduced to overcome these limitations. IGNO unifies the solution of inverse problems from both point measurements and operator-valued data without labeled training pairs. This framework encodes high-dimensional, potentially discontinuous coefficient fields into a low-dimensional latent space, which drives neural operator decoders to reconstruct both coefficients and PDE solutions. Training relies purely on physics constraints through PDE residuals, while inversion proceeds via efficient gradient-based optimization in latent space, accelerated by an a priori normalizing flow model. Across a diverse set of challenging inverse problems, including recovery of discontinuous coefficients from solution-based measurements and the EIT problem with operator-based measurements, IGNO consistently achieves accurate, stable, and scalable inversion even under severe noise. It consistently outperforms the state-of-the-art method under varying noise levels and demonstrates strong generalization to out-of-distribution targets. These results establish IGNO as a unified and powerful framework for tackling challenging inverse problems across computational science domains. Physical sciences/Mathematics and computing/Computational science Physical sciences/Mathematics and computing/Applied mathematics Inverse Problems Deep Neural Operator Deep Generative Modeling PDEs EIT Problem Full Text Additional Declarations There is NO Competing Interest. Supplementary Files supplementaryinformationrevised.pdf Supplementary Information: A unified physics-informed generative operator framework for general inverse problems Cite Share Download PDF Status: Under Review 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-7867321","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":543786869,"identity":"25c2cba5-8b02-4c58-9b28-a0f1683d1297","order_by":0,"name":"Yaohua Zang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA2UlEQVRIiWNgGAWjYHACNhiD8QGYOkCCFmaDA6RqYZMgSovB8cPHHvPUMNjzz26/Vv2xbRsD3/EGAlrOpKUb8xxjYJa4c6bsxsG22wySZwhYY3Agx0yahw3ouhs5aWAtBjcSCGg5/wao5R8DjzxQSwFYy/0HBLTcANrC28YgYXAj/RgDxBb8OhgkbzxLk5zbJ2FgeCOHWeLMuds8kmcIOIzvfPIxiTffbOzlbqQ//FBRdluO7/gB/FoUIPISQMxjAGLxEHAWA4N8A5zJ/oCg6lEwCkbBKBiZAAAsZUkfWtURigAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0003-2858-643X","institution":"Technical University of Munich","correspondingAuthor":true,"prefix":"","firstName":"Yaohua","middleName":"","lastName":"Zang","suffix":""},{"id":543786870,"identity":"b6ef40c8-6d92-4703-9328-50286e78fdc6","order_by":1,"name":"Gang Bao","email":"","orcid":"","institution":"Zhejiang University","correspondingAuthor":false,"prefix":"","firstName":"Gang","middleName":"","lastName":"Bao","suffix":""}],"badges":[],"createdAt":"2025-10-15 11:05:39","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-7867321/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-7867321/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107044023,"identity":"e737ef36-4dd0-4c41-b5a9-fb6d55b3836a","added_by":"auto","created_at":"2026-04-16 06:58:32","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3511191,"visible":true,"origin":"","legend":"Article File","description":"","filename":"manuscriptrevised.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7867321/v1_covered_0329cffb-e0d9-42e3-8b81-844351d829e5.pdf"},{"id":95884725,"identity":"e21f0e9c-7ad1-4a97-8507-f650a713aeb3","added_by":"auto","created_at":"2025-11-14 04:16:33","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":888922,"visible":true,"origin":"","legend":"Supplementary Information: A unified physics-informed generative operator framework for general inverse problems","description":"","filename":"supplementaryinformationrevised.pdf","url":"https://assets-eu.researchsquare.com/files/rs-7867321/v1/15f52a540c7e854de272f78f.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"A unified physics-informed generative operator framework for general inverse problems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nature-portfolio","isNatureJournal":true,"hasQc":false,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"","title":"Nature Portfolio","twitterHandle":"","acdcEnabled":false,"dfaEnabled":false,"editorialSystem":"ejp","reportingPortfolio":"","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Inverse Problems, Deep Neural Operator, Deep Generative Modeling, PDEs, EIT Problem","lastPublishedDoi":"10.21203/rs.3.rs-7867321/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-7867321/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Solving inverse problems governed by partial differential equations (PDEs) is central to science and engineering, yet remains challenging when measurements are sparse, noisy, or when the underlying coefficients are high-dimensional or discontinuous. Existing deep learning approaches either require extensive labeled datasets or are limited to specific measurement types, often leading to failure in such regimes and restricting their practical applicability. Here, a novel generative neural operator framework, IGNO, is introduced to overcome these limitations. IGNO unifies the solution of inverse problems from both point measurements and operator-valued data without labeled training pairs. This framework encodes high-dimensional, potentially discontinuous coefficient fields into a low-dimensional latent space, which drives neural operator decoders to reconstruct both coefficients and PDE solutions. Training relies purely on physics constraints through PDE residuals, while inversion proceeds via efficient gradient-based optimization in latent space, accelerated by an a priori normalizing flow model. Across a diverse set of challenging inverse problems, including recovery of discontinuous coefficients from solution-based measurements and the EIT problem with operator-based measurements, IGNO consistently achieves accurate, stable, and scalable inversion even under severe noise. It consistently outperforms the state-of-the-art method under varying noise levels and demonstrates strong generalization to out-of-distribution targets. 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