Learning Generalizable Neural Operators for Inverse Problems

Learning Generalizable Neural Operators for 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 Learning Generalizable Neural Operators for Inverse Problems Adam Thorpe, Stepan Tretiakov, Dibakar Sarkar, Krishna Kumar, and 1 more This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8408330/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 Inverse problems challenge existing neural operator architectures because ill-posed inverse maps violate continuity, uniqueness, and stability assumptions. We introduce B2B^-1, an inverse basis-to-basis neural operator framework that addresses this limitation. Our key innovation is to decouple function representation from the inverse map. We learn neural basis functions for the input and output spaces, then train inverse models that operate on the resulting coefficient space. This structure allows us to learn deterministic, invertible, and probabilistic models within a single framework, and to choose models based on the degree of ill-posedness. We evaluate our approach on six inverse PDE benchmarks, including two novel datasets, and compare against existing invertible neural operator baselines. We learn probabilistic models that capture uncertainty and input variability, and remain robust to measurement noise due to implicit denoising in the coefficient calculation. Our results show consistent re-simulation performance across varying levels of ill-posedness. By separating representation from inversion, our framework enables scalable surrogate models for inverse problems that generalize across instances, domains, and degrees of ill-posedness. Physical sciences/Mathematics and computing/Computational science Physical sciences/Physics/Information theory and computation Full Text Additional Declarations There is NO Competing Interest. 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. 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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-8408330","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":586054759,"identity":"4060a1ef-218f-45e4-8682-e16cca58b154","order_by":0,"name":"Adam Thorpe","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA+ElEQVRIiWNgGAWjYFCCBBBxAMqpAGJm5gYitCTAtJwBaWEkRQtjG5jEr4W/PfnpBsYfd+TMZzcfYPw6rzaavx2o5UfFNpxaJM48M7vBkPDMWObOsQRm2W3Hc2ccZmxg7DlzG7c1NxJAWg4nzpDIMWCW3HYstwGohZmxDbcW+Rvp36Ba8j8wS845ljufkBaDGzlwWxgYPzbU5G4gpMXwzJuyGwlph40lZI4ZHGY4diB3I1DLQXx+kTuevu3GB5vDchLSzQ8f/qipy513/vDBBz8q8HgfBBJAhAQDw2EehsNggQP41cMAUAvjD4Y64hSPglEwCkbBiAIAFlNjOvDquGsAAAAASUVORK5CYII=","orcid":"https://orcid.org/0000-0001-7120-0913","institution":"University of Texas at Austin","correspondingAuthor":true,"prefix":"","firstName":"Adam","middleName":"","lastName":"Thorpe","suffix":""},{"id":586054760,"identity":"06c53539-e0a4-4fc8-889a-95124c0ae997","order_by":1,"name":"Stepan Tretiakov","email":"","orcid":"","institution":"University of Texas at Austin","correspondingAuthor":false,"prefix":"","firstName":"Stepan","middleName":"","lastName":"Tretiakov","suffix":""},{"id":586054761,"identity":"0e07e91d-1b33-47de-9e1d-7d9384f91c84","order_by":2,"name":"Dibakar Sarkar","email":"","orcid":"","institution":"Johns Hopkins University","correspondingAuthor":false,"prefix":"","firstName":"Dibakar","middleName":"","lastName":"Sarkar","suffix":""},{"id":586054762,"identity":"df44f985-7cd2-4c9f-bafd-3720305ee1dd","order_by":3,"name":"Krishna Kumar","email":"","orcid":"","institution":"University of Texas at Austin","correspondingAuthor":false,"prefix":"","firstName":"Krishna","middleName":"","lastName":"Kumar","suffix":""},{"id":586054763,"identity":"3e511b74-88b3-4a09-8219-afe27076269e","order_by":4,"name":"Ufuk Topcu","email":"","orcid":"","institution":"University of Texas","correspondingAuthor":false,"prefix":"","firstName":"Ufuk","middleName":"","lastName":"Topcu","suffix":""}],"badges":[],"createdAt":"2025-12-19 21:55:06","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8408330/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8408330/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106404650,"identity":"721dc3f7-6ebd-4564-acd2-598b9341e1d9","added_by":"auto","created_at":"2026-04-08 09:16:27","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1682904,"visible":true,"origin":"","legend":"Article File","description":"","filename":"NMIInverseNeuralOperators17.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8408330/v1_covered_08f1a58e-7521-4ba8-93cc-9ad302f54175.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"Learning Generalizable Neural Operators for Inverse Problems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"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":"","lastPublishedDoi":"10.21203/rs.3.rs-8408330/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8408330/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Inverse problems challenge existing neural operator architectures because ill-posed inverse maps violate continuity, uniqueness, and stability assumptions. 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