GREENKAN: Structured Spatiotemporal Green-Function Neural Operators with Interpretable Kernel Decomposition.

GREENKAN: Structured Spatiotemporal Green-Function Neural Operators with Interpretable Kernel Decomposition. | 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 GREENKAN: Structured Spatiotemporal Green-Function Neural Operators with Interpretable Kernel Decomposition. Mohammed Ward This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9212738/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 Partial differential equations (PDEs) are central to modeling physical systems, and neural operators provide a data-driven framework for learning mappings between function spaces. Most existing neural operators rely on implicit spectral or feature-space representations, which offer limited interpretability of the learned solution structure. We introduce GREENKAN, a structured neural operator for one-dimensional time-dependent linear PDEs that parameterizes solutions through a separable spatiotemporal kernel expansion inspired by Green-function representations. Building on the functional parameterization philosophy of Kolmogorov– Arnold Networks (KAN) [9], the model learns explicit families of spatial and temporal kernels with controllable scale, frequency, and decay characteristics. A hypernetwork [14] generates synthesis coefficients conditioned on the input problem, while a gated symmetric amplification mechanism promotes stable training and mitigates mode collapse. By explicitly modeling kernel structure, GREENKAN enables direct inspection of learned basis functions and their temporal dynamics, yielding physically interpretable internal representations. This structured formulation provides a principled and transparent alternative to fully implicit neural operator architectures. This work provides a structured foundation for interpretable operator learning, with a natural path toward extensions to nonlinear and higher-dimensional PDEs. Artificial Intelligence and Machine Learning Physics-Informed AI Differential Equations Kolmogorov– Arnold Networks (KAN) Green's Functions 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. 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