A Rotationally Equivariant Graph Neural Network Framework for the Inverse Design of Metastable Multi-Principal Element Alloys: A Rigorous Computational and Mathematical Approach

A Rotationally Equivariant Graph Neural Network Framework for the Inverse Design of Metastable Multi-Principal Element Alloys: A Rigorous Computational and Mathematical Approach | 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 A Rotationally Equivariant Graph Neural Network Framework for the Inverse Design of Metastable Multi-Principal Element Alloys: A Rigorous Computational and Mathematical Approach Satish Prajapati This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9451117/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 The discovery of multi-principal element alloys (MPEAs) with exceptional mechanical properties repre sents one of the grand challenges in modern materials science. The combinatorial explosion of compo sition space—exceeding 1012 candidates for quinary systems—renders exhaustive experimental or com putational screening impossible. Here we present a mathematically rigorous computational framework that integrates four components to address this challenge: (i) high-throughput density functional the ory (DFT) calculations using the SCAN meta-GGA functional with frozen-phonon vibrational entropy and stochastic metastable sampling; (ii) a rotationally equivariant graph neural network (RE-GNN) that provably satisfies exact E(3) equivariance (Theorem 1); (iii) a deep ensemble uncertainty quan tification scheme with provable calibration guarantees (Theorem 2); and (iv) a genetic algorithm with proven almost-sure convergence to the global optimum (Theorem 3). Applied to the Nb-Mo-Ta-W-V system, our method reduces the search space from > 1012 to fewer than 103 candidate calculations, discovering three previously unreported metastable MPEAs with predicted yield strengths exceeding 2.55 GPa and elongations to failure exceeding 14.8%. Phonon calculations confirm dynamical stability, and temperature-dependent free energy analysis reveals thermodynamic stabilization above 400 K. This framework establishes a mathematically rigorous foundation for machine-learning-accelerated metastable alloy discovery. All code, data, and supplementary information are provided for full reproducibility. Materials Engineering Materials Theory and Modeling Multi-principal element alloys high-entropy alloys equivariant graph neural networks E(3) equivariance uncertainty quantification deep ensemble active learning genetic algorithm inverse design metastable materials density functional theory SCAN meta-GGA refractory alloys Nb-Mo-Ta-W-V system phonon calculations free energy stabilization materials discovery computational materials science machine learning interatomic potentials Clebsch-Gordan decomposition spherical harmonics Wigner D-matrices Pareto optimization strength-to-weight ratio yield strength prediction ductility convex hull vibrational entropy Full Text Additional Declarations The authors declare no competing interests. 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. 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-9451117","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":625103451,"identity":"b40f1d84-f165-454b-9740-ecab6a422aab","order_by":0,"name":"Satish Prajapati","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABI0lEQVRIiWNgGAWjYBACNgglwWCAEGM+ABKRwa8lAa4FRLElgLTw4LcrgQFZCw+YjVMLH/vp5A8/f1jYmzOwX/zwoeYPA3/7mc+vbtRY8DCwHz66AZvDeHK3SfYkSCTubOAplpxxzIBB4kzuNuucY0CH8aSl3cDql9xtDDwJEgkGB3jSmHkbgG66wbvNOIcNqEWCxwyrFv63mz/+SZCwB2v5C9Qif4PnmXHOPzxaJHI3SANtYdxwgP0YMyNQi8ENHubHuW34tLzdJi2TJpG44TAPs2TPMWMewzNpZsy5fRI8bDj8It+fu/njG5s6e4Pj7Q8//KiRk5M7fvjx55xvdXL87IePYdOCAMyI6GCTgIQMQcD+AK77A2HVo2AUjIJRMIIAACioWpYTNplgAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0009-0006-3801-1137","institution":"Government College of Engineering and Ceramic Technology, Kolkata, India","correspondingAuthor":true,"prefix":"","firstName":"Satish","middleName":"","lastName":"Prajapati","suffix":""}],"badges":[],"createdAt":"2026-04-17 16:06:43","currentVersionCode":1,"declarations":{"humanSubjects":false,"vertebrateSubjects":true,"conflictsOfInterestStatement":false,"humanSubjectEthicalGuidelines":false,"humanSubjectConsent":false,"humanSubjectClinicalTrial":false,"humanSubjectCaseReport":false,"vertebrateSubjectEthicalGuidelines":true},"doi":"10.21203/rs.3.rs-9451117/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9451117/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":107484527,"identity":"b237be6c-a352-4d6c-9f9d-3ba905a499aa","added_by":"auto","created_at":"2026-04-22 02:32:19","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":6171290,"visible":true,"origin":"","legend":"","description":"","filename":"V.2ARotationallyEquivariantGraphNeuralNetworkFramework.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9451117/v1_covered_69eda43a-2b87-4bbc-b431-01102b6a571f.pdf"}],"financialInterests":"The authors declare no competing interests.","formattedTitle":"\u003cp\u003eA Rotationally Equivariant Graph Neural Network Framework for the Inverse Design of Metastable Multi-Principal Element Alloys: A Rigorous Computational and Mathematical Approach\u003c/p\u003e","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":"Multi-principal element alloys, high-entropy alloys, equivariant graph neural networks, E(3) equivariance, uncertainty quantification, deep ensemble, active learning, genetic algorithm, inverse design, metastable materials, density functional theory, SCAN meta-GGA, refractory alloys, Nb-Mo-Ta-W-V system, phonon calculations, free energy stabilization, materials discovery, computational materials science, machine learning interatomic potentials, Clebsch-Gordan decomposition, spherical harmonics, Wigner D-matrices, Pareto optimization, strength-to-weight ratio, yield strength prediction, ductility, convex hull, vibrational entropy","lastPublishedDoi":"10.21203/rs.3.rs-9451117/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9451117/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThe discovery of multi-principal element alloys (MPEAs) with exceptional mechanical properties repre sents one of the grand challenges in modern materials science. The combinatorial explosion of compo sition space—exceeding 1012 candidates for quinary systems—renders exhaustive experimental or com putational screening impossible. Here we present a mathematically rigorous computational framework that integrates four components to address this challenge: (i) high-throughput density functional the ory (DFT) calculations using the SCAN meta-GGA functional with frozen-phonon vibrational entropy and stochastic metastable sampling; (ii) a rotationally equivariant graph neural network (RE-GNN) that provably satisfies exact E(3) equivariance (Theorem 1); (iii) a deep ensemble uncertainty quan tification scheme with provable calibration guarantees (Theorem 2); and (iv) a genetic algorithm with proven almost-sure convergence to the global optimum (Theorem 3). 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