[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-328193-105":59,"doc-detail-328193-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","intrinsic-greens-learning-supervised-learning-on-manifolds-via-inverse-pde","INTRINSIC GREEN ’S LEARNING: SUPERVISED LEARNING ON MANIFOLDS VIA INVERSE PDE","","Intrinsic Green’s Learning (IGL) models a target function on a manifold as the solution of a linear PDE, where the PDE source term is learned from data. Instead of approximating the target directly, IGL learns a source and integrates it with a Green’s kernel. An encoder finds an intrinsic low-dimensional coordinate chart enabling low-rank tensor decompositions and turning a high-dimensional integral into independent one-dimensional integrals with cost linear in intrinsic dimension. A two-stage algorithm separates coordinate discovery from source fitting, using a near-convex linear solve to avoid collapse, and learnable gates recover the manifold’s intrinsic dimension. Experiments on synthetic manifolds and MNIST show near-optimal classification and automatic intrinsic-dimension recovery.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/intrinsic-greens-learning-supervised-learning-on-manifolds-via-inverse-pde/328193/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/intrinsic-greens-learning-supervised-learning-on-manifolds-via-inverse-pde/328193.png","ImageObject",300,407,{"name":92,"@type":93},"Aurelia","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-13","2026-09-21",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What problem does Intrinsic Green’s Learning (IGL) address in supervised learning?","Question",{"text":113,"@type":114},"IGL addresses supervised learning when data lie on an unknown low-dimensional intrinsic manifold, aiming to learn the target function while exploiting manifold structure.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How does IGL represent the target function?",{"text":118,"@type":114},"IGL parameterizes the target function as the solution of a linear PDE, where the source term is learned from data and integrated against a Green’s kernel.",{"name":120,"@type":111,"acceptedAnswer":121},"What are the key components of IGL’s two-stage algorithm?",{"text":122,"@type":114},"Stage 1 uses an encoder to discover an intrinsic coordinate chart where the source and operator decompose into low-rank tensors. Stage 2 fits the source weights (and kernel scales) using the discovered coordinates via fast, nearly convex linear solving.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},328193,1791071901,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":56,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},1099514068365,"https://ap-avatar.wpscdn.com/avatar/10000253d8d9f28188e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1794309523707217728","arXiv :2607 .07034v 1 [ cs .LG] 8 Jul 2026  \nINTRINSIC GREEN ’S LEARNING: SUPERVISED LEARNING ON MANIFOLDS VIA INVERSE PDE  \nAlexandre Quemy  \nHother Labs  \n[alexandre@hother.io](alexandre@hother.io)  \nABSTRACT  \nWe introduce Intrinsic Green’s Learning (IGL), a framework that models a target function on a manifold as the solution to a linear PDE whose source term is learned from data. Rather than approximating the target directly, IGL learns a source and integrates it against a Green’s kernel. An encoder discovers a lowdimensional coordinate chart on the manifold where both the source and the kernel decompose as low-rank tensors, collapsing a high-dimensional integral into independent one-dimensional integrals with cost linear in the intrinsic dimension. A two-stage algorithm separates coordinate discovery from source fitting, a nearconvex linear solve, preventing the dimensional collapse of joint training. Learnable gates on each coordinate automatically discover the intrinsic dimension of the manifold. We validate IGL on synthetic manifolds and on MNIST, where it simultaneously achieves near-optimal classification and automatic recovery of the intrinsic dimension.  \n1 INTRODUCTION  \nConsider the standard supervised learning problem: given observations {(x(n), y (n))}, learn a function u that maps inputs to outputs. A commonly accepted hypothesis is that data lies on a lowdimensional (intrinsic) manifold M ⊂ RD with intrinsic dimension d ≪ D. Knowing the manifold structure has many potential benefits: it can reduce sample complexity, improve generalization, and enable interpretability(Pope et al., 2021; Ansuini et al., 2019; Mao et al., 2024) . However, learning on manifolds is challenging: the manifold is unknown, the ambient dimension is high, and the topology may be complex.  \nIn this paper, we propose Intrinsic Green’s Learning (IGL), a new approach where we do not try to discover the intrinsic manifold but a coordinate chart on a low dimensional manifold that reveals the low-rank structure of the source term and operator of a linear PDE. This is a weaker goal than discovering the manifold itself, but it provides a powerful structural prior that enables tractable learning and generalization.  \nRather than modeling u directly, we parameterize it as the solution to a linear partial differential equation Lu = f where L is a differential operator and f is a learned source term. This isan architectural choice, not a physical claim: the PDE assumption provides (a) a decomposable representation via tensor products of the source and Green’s function,(b) a smoothing inductive bias since integration is a stable operation unlike differentiation (Raissi et al., 2019), and (c) linearity in source weights, enabling the separation of geometry from fitting that we exploit below. The operator L provides a principled inductive bias: smoothness for the Laplacian, locality for Helmholtz, longrange correlations for fractional operators.  \nThe two-stage algorithm. The central contribution of this paper is a two-stage algorithm that separates coordinate discovery from source fitting (Figure 1):  \n• Stage 1 (Coordinate Discovery): An encoder Ψ : RD → Rd discovers a coordinate chart where the source f and operator L decompose as low-rank tensors such that solving the EDP is essentially equivalent to d 1D integrals.  \nFigure 1: IGL pipeline as a two-stage algorithm. Stage 1 (coordinate discovery): the encoder Ψ maps data from a manifold M ⊂ RD (left) to intrinsic coordinates ξ ∈ Rd where the source decomposes. The tensor source ˆf(ζ) = Pr wr Qj ϕr,j (ζj ) is defined by anchor points µr,j (green dots) . Green’s convolution G ∗ ˆfyields the solution u(ξ) . Stage 2 (source fitting): source weights wr are solved via gradient descent or least squares (Appendix C.1) .  \n• Stage 2 (Source Fitting): Given coordinates from Stage 1, solve for source weights and kernel scales. The factorized integral (Eq. 4) is linear, making this a nearly convex problem wit","cbCaiuIi7I6SE6hU","https://ap.wps.com/l/cbCaiuIi7I6SE6hU","pdf",4002795,"English","# Abstract\n# Introduction\n# Related Work","[{\"question\":\"What problem does Intrinsic Green’s Learning (IGL) address in supervised learning?\",\"answer\":\"IGL addresses supervised learning when data lie on an unknown low-dimensional intrinsic manifold, aiming to learn the target function while exploiting manifold structure.\"},{\"question\":\"How does IGL represent the target function?\",\"answer\":\"IGL parameterizes the target function as the solution of a linear PDE, where the source term is learned from data and integrated against a Green’s kernel.\"},{\"question\":\"What are the key components of IGL’s two-stage algorithm?\",\"answer\":\"Stage 1 uses an encoder to discover an intrinsic coordinate chart where the source and operator decompose into low-rank tensors. Stage 2 fits the source weights (and kernel scales) using the discovered coordinates via fast, nearly convex linear solving.\"}]","INTRINSIC GREEN ’S LEARNING: SUPERVISED LEARNING ON MANIFOLDS VIA INVERSE PDE | PDF",1789978811,48]