HOMO-PINN: Hyperparameter Optimization of a Multi-Output Physics-Informed Neural Network

HOMO-PINN: Hyperparameter Optimization of a Multi-Output Physics-Informed Neural Network | 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 HOMO-PINN: Hyperparameter Optimization of a Multi-Output Physics-Informed Neural Network Salvatore Cuomo, Maria Pia de Rosa, Laura Pompameo, Alexander Litvienko This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6880919/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 10 Oct, 2025 Read the published version in Operations Research Forum → Version 1 posted 9 You are reading this latest preprint version Abstract The good choice of hyperparameters is crucial for the successful application of Deep Learning (DL) networks in order to find accurate solutions or the best parameter in solving Partial Differential Equations (PDEs), that are sensitive to errors in coefficient estimation. For this purpose, Hyperparameter Optimization of Multi-Output Physics-Informed Neural Networks (HOMO-PINNs) is based on the optimal search of PINN hyperparameters for solving PDEs with uncertain coefficients in the Uncertainty Quantification (UQ) field. By testing this novel methodology on different PDEs, the relationship between activation functions, the number of output neurons, and the degree of coefficient uncertainty can be observed. The experimental results show that adding output neurons to the Neural Network (NN) even if a theoretically incorrect activation function is chosen, keeps the predicted solution accurate. Physics-Informed Neural Network Numerical methods PDE Uncertainty Quantification UQ Hyperparameter optimization Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 10 Oct, 2025 Read the published version in Operations Research Forum → Version 1 posted Editorial decision: Revision requested 12 Aug, 2025 Reviews received at journal 12 Aug, 2025 Reviews received at journal 11 Jul, 2025 Reviewers agreed at journal 30 Jun, 2025 Reviewers agreed at journal 24 Jun, 2025 Reviewers invited by journal 18 Jun, 2025 Editor assigned by journal 13 Jun, 2025 Submission checks completed at journal 13 Jun, 2025 First submitted to journal 12 Jun, 2025 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-6880919","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":473091902,"identity":"5100de15-4d33-4f8a-8020-e506663ee969","order_by":0,"name":"Salvatore Cuomo","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA8ElEQVRIiWNgGAWjYFAC5gYUrhwDAyNIRAKPFpCCBATXGKYFjx40LYkwS3FqkW8/2Cbx84dNPj8D8zOJjzk26RtuN7c9+MFgUYdLi8GZxDbJnoQ0y5kNbGaSM7el5W64c7DdsAePwwwYEpsNeBIOGxgc4GGT5t12OHfDjcQ2CR48WuT7HzYb/gFqsYdqSTcAapH8g8/7NxIbH4NtYYBoSQBpkcZni8GNh42PZdLSDCQOsxlbAv1iOBOkRcZAQrIBp8OSDxx8Y2NjwN/e/PDGx2028nw30p9Jvqmo48fpMDhgRrWdsIZRMApGwSgYBbgBAOLCUNGP0bUBAAAAAElFTkSuQmCC","orcid":"","institution":"University of Naples Federico II","correspondingAuthor":true,"prefix":"","firstName":"Salvatore","middleName":"","lastName":"Cuomo","suffix":""},{"id":473091903,"identity":"04a37062-ae11-48e0-baec-884f29e01cab","order_by":1,"name":"Maria Pia de Rosa","email":"","orcid":"","institution":"University of Naples Federico II","correspondingAuthor":false,"prefix":"","firstName":"Maria","middleName":"Pia","lastName":"de Rosa","suffix":""},{"id":473091904,"identity":"2a26bec8-cb9e-4daa-b8de-d49b47da1a15","order_by":2,"name":"Laura Pompameo","email":"","orcid":"","institution":"University of Naples Federico II","correspondingAuthor":false,"prefix":"","firstName":"Laura","middleName":"","lastName":"Pompameo","suffix":""},{"id":473091905,"identity":"878769ec-dee0-454a-8b55-321a16e0395c","order_by":3,"name":"Alexander Litvienko","email":"","orcid":"","institution":"RWTH Aachen University","correspondingAuthor":false,"prefix":"","firstName":"Alexander","middleName":"","lastName":"Litvienko","suffix":""}],"badges":[],"createdAt":"2025-06-12 13:38:32","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6880919/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6880919/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s43069-025-00561-7","type":"published","date":"2025-10-10T15:57:50+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":93419739,"identity":"58d9524b-d653-414a-b381-771cc1f4c3f9","added_by":"auto","created_at":"2025-10-13 16:06:47","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3918934,"visible":true,"origin":"","legend":"","description":"","filename":"Archivio.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6880919/v1_covered_10110ba2-e6ab-48c5-bec0-e20822591221.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"HOMO-PINN: Hyperparameter Optimization of a Multi-Output Physics-Informed Neural Network","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"operations-research-forum","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"","sideBox":"Learn more about [Operations Research Forum](https://link.springer.com/journal/43069)","snPcode":"43069","submissionUrl":"https://submission.nature.com/new-submission/43069/3","title":"Operations Research Forum","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Physics-Informed Neural Network, Numerical methods, PDE, Uncertainty Quantification, UQ, Hyperparameter optimization","lastPublishedDoi":"10.21203/rs.3.rs-6880919/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6880919/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"The good choice of hyperparameters is crucial for the successful application of Deep Learning (DL) networks in order to find accurate solutions or the best parameter in solving Partial Differential Equations (PDEs), that are sensitive to errors in coefficient estimation. 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