LegONet: Plug-and-Play Structure-Preserving Neural Operator Blocks for Compositional PDE Learning

LegONet: Plug-and-Play Structure-Preserving Neural Operator Blocks for Compositional PDE Learning | 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 LegONet: Plug-and-Play Structure-Preserving Neural Operator Blocks for Compositional PDE Learning Guang Lin, Jiahao Zhang, Yueqi Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9067223/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 Learned PDE solvers are often trained as monolithic surrogates for a specific equation, boundary condition and discretization. This makes them difficult to reuse when mechanisms change and it can limit stability under long-horizon rollout. We introduce Lego-like Operator Network (LegONet), a compositional framework that builds PDE solvers from plug-and-play, structure-preserving operator blocks defined on shared boundary-adapted spectral representations. LegONet separates boundary handling from mechanism learning, satisfying boundary conditions by construction. It also separates mechanism learning from time integration, enabling pretrained blocks to be assembled into new solvers without retraining. We also derive a finite-horizon error decomposition that separates block mismatch from splitting error and provides mechanism-level diagnostics for long-horizon predictions. Across ten time-dependent PDEs, LegONet delivers accurate closed loop rollouts with improved stability under cross-PDE recombination and boundary reconfiguration. More broadly, this modular formulation suggests a path from task-specific neural solvers towards plug-and-play operator libraries for scientific computing. Physical sciences/Mathematics and computing/Computational science Physical sciences/Mathematics and computing/Applied mathematics Operator Learning Neural Operator Structure-Preserving Learning Operator Splitting Compositional Modeling Trajectory-Free Training Full Text Additional Declarations There is NO Competing Interest. Supplementary Files SupplementaryInformation2.pdf Supplementary Information 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. 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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-9067223","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":624818238,"identity":"ce6412ac-6d9b-4b0c-890b-9cdc318879ac","order_by":0,"name":"Guang Lin","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA30lEQVRIiWNgGAWjYNCCAgkGfhDNA+YlEKPFQIJBsoFELUB0gFgt8hHJzx5+MbDIM77dY/bhbZsNAz97jgFeLYY30syNZQwkis3unDGeObctjUGy5w0BLTMSzKQlDCQSt93IMWbmbTvMYHCDkC0z0r+BtWyeAdbyn8GekBZ5iRwzyQ9ALRskwFoOAEOPgBYDnjdl0kBliTPuHCtmnHMumUfizLMC/La0p2+T/FFRl9g/u3kzw5syOzn+9uQN+G05wMDADI4OCYgAD17lYFsaGBgYfyBpGQWjYBSMglGAAQDNa0G6zSEXwAAAAABJRU5ErkJggg==","orcid":"https://orcid.org/0000-0002-0976-1987","institution":"Purdue University","correspondingAuthor":true,"prefix":"","firstName":"Guang","middleName":"","lastName":"Lin","suffix":""},{"id":624818239,"identity":"ebdfb191-324a-4053-ac68-885f264cd1d1","order_by":1,"name":"Jiahao Zhang","email":"","orcid":"","institution":"Purdue University","correspondingAuthor":false,"prefix":"","firstName":"Jiahao","middleName":"","lastName":"Zhang","suffix":""},{"id":624818240,"identity":"9942e7fa-b61b-4431-bcda-cedd9fbab292","order_by":2,"name":"Yueqi Wang","email":"","orcid":"","institution":"Purdue University","correspondingAuthor":false,"prefix":"","firstName":"Yueqi","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2026-03-09 01:35:15","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9067223/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9067223/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107705210,"identity":"7b0ba0b1-41ce-46d9-bd7b-82d95cf5de75","added_by":"auto","created_at":"2026-04-24 09:09:37","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":6302379,"visible":true,"origin":"","legend":"Article File","description":"","filename":"RevisedNMI.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9067223/v1_covered_8b2e048f-0864-4b22-92ff-a8e120b82891.pdf"},{"id":107282719,"identity":"41e8e9b4-6f85-48b2-b7f5-7e9e4fb714ae","added_by":"auto","created_at":"2026-04-20 02:48:35","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"supplement","size":391361,"visible":true,"origin":"","legend":"Supplementary Information","description":"","filename":"SupplementaryInformation2.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9067223/v1/0bd66a98c957556e532fe159.pdf"}],"financialInterests":"There is \u003cb\u003eNO\u003c/b\u003e Competing Interest.","formattedTitle":"LegONet: Plug-and-Play Structure-Preserving Neural Operator Blocks for Compositional PDE Learning","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":"Operator Learning, Neural Operator, Structure-Preserving Learning, Operator Splitting, Compositional Modeling, Trajectory-Free Training","lastPublishedDoi":"10.21203/rs.3.rs-9067223/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9067223/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Learned PDE solvers are often trained as monolithic surrogates for a specific equation, boundary condition and discretization. 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