Enabling PINNs for stiff moving-boundary PDEs: Locking-point prediction in superheated steam drying

Enabling PINNs for stiff moving-boundary PDEs: Locking-point prediction in superheated steam drying | 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 Enabling PINNs for stiff moving-boundary PDEs: Locking-point prediction in superheated steam drying Narjes Malekjani, Andreas Bück, Evangelos Tsotsas, Abdolreza Kharaghani This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8828013/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 4 You are reading this latest preprint version Abstract Physics-informed neural networks (PINNs) typically fail in solving stiff PDEs with moving boundary conditions such as in convection-dominated droplet drying with shrinking domains and in Stefan-type phase-change problems. This study introduces a step-wise approach (scaled logarithmically transformed PINN, LT-PINN) to resolve this issue. The approach is applied to the process of nanosuspension droplet drying in superheated steam exhibiting severe stiffness due to exponential spatial concentration increase and high drying rates. Scaled LT-PINN introduces two synergistic innovations: 1) a logarithmic state-space transformation that compresses the dynamic range and linearizes gradients, and 2) an inverse Péclet -number boundary loss scaling, ensuring a balanced training signal across all stiffness regimes. This approach eliminates the loss imbalance that causes error degradation at high Péclet numbers (∆T = 100 K) reducing the mean relative L 2 error from 97.05 ± 1.22% (Baseline) and 5.44 ± 1.49% (LT-PINN) to 2.59 ± 0.77% (Scaled LT-PINN). Scaled LT-PINN was validated against Crank-Nicolson benchmarks across a wide range of superheating temperatures (∆T = 10–100 K) and achieved a mean relative L 2 error below 2.97 ± 0.49% and a locking time error under 8.52 ± 3.01% even at the most extreme drying rates compared to the reference Crank-Nicolson solution. This framework offers a robust approach for solving stiff PDEs with steep gradients and moving boundaries. Physical sciences/Engineering Physical sciences/Mathematics and computing Physical sciences/Physics Advection-diffusion loss normalization physics-informed neural networks stiffness stabilization spectral bias Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 12 Feb, 2026 Editor assigned by journal 11 Feb, 2026 Submission checks completed at journal 11 Feb, 2026 First submitted to journal 09 Feb, 2026 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-8828013","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Article","associatedPublications":[],"authors":[{"id":588104281,"identity":"a323c20f-3523-43c0-90d6-98e5d82f0d92","order_by":0,"name":"Narjes Malekjani","email":"data:image/png;base64,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","orcid":"","institution":"Otto von Guericke University Magdeburg","correspondingAuthor":true,"prefix":"","firstName":"Narjes","middleName":"","lastName":"Malekjani","suffix":""},{"id":588104282,"identity":"1f5a7494-ae12-4d7d-813f-5b4fe848b3c9","order_by":1,"name":"Andreas Bück","email":"","orcid":"","institution":"Otto von Guericke University Magdeburg","correspondingAuthor":false,"prefix":"","firstName":"Andreas","middleName":"","lastName":"Bück","suffix":""},{"id":588104283,"identity":"3e00a116-e224-4bf3-af90-d15c6b343ea4","order_by":2,"name":"Evangelos Tsotsas","email":"","orcid":"","institution":"Otto von Guericke University Magdeburg","correspondingAuthor":false,"prefix":"","firstName":"Evangelos","middleName":"","lastName":"Tsotsas","suffix":""},{"id":588104284,"identity":"295a6030-14fc-4ade-a74f-f85a4920efca","order_by":3,"name":"Abdolreza Kharaghani","email":"","orcid":"","institution":"Otto von Guericke University Magdeburg","correspondingAuthor":false,"prefix":"","firstName":"Abdolreza","middleName":"","lastName":"Kharaghani","suffix":""}],"badges":[],"createdAt":"2026-02-09 08:53:29","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8828013/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8828013/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":102298042,"identity":"f6c84771-c78f-4c3c-88a3-0980c8fb2bb7","added_by":"auto","created_at":"2026-02-10 10:30:10","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1252588,"visible":true,"origin":"","legend":"","description":"","filename":"PINNforsupeheatedsteamdryingMalekjaniVABET.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8828013/v1_covered_d1978ba1-1c03-41a7-8f9f-089f7cffd335.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Enabling PINNs for stiff moving-boundary PDEs: Locking-point prediction in superheated steam drying","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":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"Advection-diffusion, loss normalization, physics-informed neural networks, stiffness stabilization, spectral bias","lastPublishedDoi":"10.21203/rs.3.rs-8828013/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8828013/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003ePhysics-informed neural networks (PINNs) typically fail in solving stiff PDEs with moving boundary conditions such as in convection-dominated droplet drying with shrinking domains and in Stefan-type phase-change problems. This study introduces a step-wise approach (scaled logarithmically transformed PINN, LT-PINN) to resolve this issue. The approach is applied to the process of nanosuspension droplet drying in superheated steam exhibiting severe stiffness due to exponential spatial concentration increase and high drying rates. Scaled LT-PINN introduces two synergistic innovations: 1) a logarithmic state-space transformation that compresses the dynamic range and linearizes gradients, and 2) an inverse P\u0026eacute;clet -number boundary loss scaling, ensuring a balanced training signal across all stiffness regimes. This approach eliminates the loss imbalance that causes error degradation at high P\u0026eacute;clet numbers (∆T\u0026thinsp;=\u0026thinsp;100 K) reducing the mean relative L\u003csub\u003e2\u003c/sub\u003e error from 97.05\u0026thinsp;\u0026plusmn;\u0026thinsp;1.22% (Baseline) and 5.44\u0026thinsp;\u0026plusmn;\u0026thinsp;1.49% (LT-PINN) to 2.59\u0026thinsp;\u0026plusmn;\u0026thinsp;0.77% (Scaled LT-PINN). Scaled LT-PINN was validated against Crank-Nicolson benchmarks across a wide range of superheating temperatures (∆T\u0026thinsp;=\u0026thinsp;10\u0026ndash;100 K) and achieved a mean relative L\u003csub\u003e2\u003c/sub\u003e error below 2.97\u0026thinsp;\u0026plusmn;\u0026thinsp;0.49% and a locking time error under 8.52\u0026thinsp;\u0026plusmn;\u0026thinsp;3.01% even at the most extreme drying rates compared to the reference Crank-Nicolson solution. This framework offers a robust approach for solving stiff PDEs with steep gradients and moving boundaries.\u003c/p\u003e","manuscriptTitle":"Enabling PINNs for stiff moving-boundary PDEs: Locking-point prediction in superheated steam drying","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-02-10 09:13:30","doi":"10.21203/rs.3.rs-8828013/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-02-12T11:30:47+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-02-11T14:22:58+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2026-02-11T14:20:42+00:00","index":"","fulltext":""},{"type":"submitted","content":"Scientific Reports","date":"2026-02-09T08:26:35+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"scientific-reports","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"scirep","sideBox":"Learn more about [Scientific Reports](http://www.nature.com/srep/)","snPcode":"","submissionUrl":"","title":"Scientific Reports","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Scientific Reports","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"a4855ec7-859d-443c-aa13-92aa3e6b703f","owner":[],"postedDate":"February 10th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"in-revision","subjectAreas":[{"id":62560455,"name":"Physical sciences/Engineering"},{"id":62560456,"name":"Physical sciences/Mathematics and computing"},{"id":62560457,"name":"Physical sciences/Physics"}],"tags":[],"updatedAt":"2026-04-07T05:23:15+00:00","versionOfRecord":[],"versionCreatedAt":"2026-02-10 09:13:30","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8828013","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8828013","identity":"rs-8828013","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}