Deep Wavelet Diffeomorphism: A New Approach to Manifold Regularization for PINN Training

Deep Wavelet Diffeomorphism: A New Approach to Manifold Regularization for PINN Training | 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 Deep Wavelet Diffeomorphism: A New Approach to Manifold Regularization for PINN Training Shijiao Gao, Hanying Gao, Carlo Cattani, Shuli Mei This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8582928/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 Physics-Informed Neural Networks (PINNs) are a mesh-free, data-driven approach for solving partial differential equations (PDEs). However, their accuracy in capturing strong-gradient regions (such as shock waves) depends on dense, uniform collocation points, which can lead to convergence issues and susceptibility to local optima. Traditional adaptive wavelet solvers excel at identifying discontinuous regions but suffer from time-step constraints and require cumbersome threshold tuning. To address these challenges, this paper proposes a Deep Wavelet Diffeomorphism (DWD) framework that integrates the strengths of both methods through a decoupled mechanism involving manifold regularization and low-dimensional training. Leveraging interpolating multiresolution analysis, DWD constructs a deep wavelet diffeomorphism that maps distorted physical-space training manifolds into smooth computational-space manifolds, while generating an optimal non-uniform collocation set---dense in feature-rich regions and sparse in smooth regions. This manifold mapping discretizes high-dimensional PDEs into a low-dimensional collocation system, which is then input into a standard PINN for efficient and stable optimization. Experimental results demonstrate that DWD achieves accuracy comparable to uniform super-resolution PINNs, while reducing collocation points by an order of magnitude and shortening training time by 5 to 8 times. It outperforms pure wavelet solvers and traditional PINNs in both accuracy and robustness. Concise and compatible with existing deep learning pipelines, DWD offers a data-efficient, low-dimensional discretization approach for high-dimensional PDEs and inverse problems in physics-informed learning. Physics-Informed Neural Networks Deep Wavelet Diffeomorphism Partial Differential Equations Low-dimensional Discretization Full Text Additional Declarations No competing interests reported. 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-8582928","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":577653771,"identity":"b0950bda-9d90-4eb6-a6a6-af53b9a18042","order_by":0,"name":"Shijiao Gao","email":"","orcid":"","institution":"China Agricultural University","correspondingAuthor":false,"prefix":"","firstName":"Shijiao","middleName":"","lastName":"Gao","suffix":""},{"id":577653772,"identity":"f613102d-91dc-4652-a973-2c1f219a0e19","order_by":1,"name":"Hanying Gao","email":"","orcid":"","institution":"China Agricultural 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