[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82765-en":3,"doc-seo-82765-105":30,"detail-sidebar-cat-0-en-105":83},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},82765,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS)","Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS) presents a supervised neural representation-learning framework that injects a spectral inductive bias through a modified Dirichlet energy. Scalar coordinate maps are trained using empirical Gram orthogonality and a supervised linear readout with cross-entropy. A learnable metric A(x)=Λ(x)U(x) transforms input gradients: Λ(x) governs anisotropic scaling, while U(x) is parameterized by structured products of Givens rotations. Experiments on synthetic, tabular, and image benchmarks analyze identity, diagonal, and rotation-scaling metrics, comparing against classical baselines and NeuralEF, emphasizing optimization stability and validation on explicit operator eigenproblems.","PIEFS: Physics-Informed Eigenfunction Features with Learnable Scaling  \nVarvara Nazarenko 1 Timur Lidzhiev 1 Alexander Tarakanov 1 2  \narXiv :2607 .03692v2 [ cs .LG] 7 Jul 2026  \nAbstract  \nSpectral methods are widely used to construct representations from the geometry of data, but they often rely on a fixed kernel, graph Laplacian, or manually selected feature scaling. We propose Physics-Informed Eigenfunction Features with Learnable Scaling (PIEFS), a supervised neural representation-learning framework with a spectral inductive bias, based on a modified Dirichlet energy. In PIEFS, scalar coordinate maps are trained under empirical Gram orthogonality, a supervised linear readout, and a Dirichlet penalty in which the input gradient is transformed by a learnable metric A (x) = Λ(x)U (x) . The diagonal factor Λ(x) controls anisotropic scaling, while the orthogonal factor U (x) is parameterized by a structured product of Givens rotations.  \nThis construction yields task-adaptive Dirichletregularized coordinates rather than eigenfunctions of a fixed supervision-independent operator. Experiments on synthetic, tabular, and image-based benchmarks study the effect of identity, diagonal, and rotation-scaling metrics, and compare the resulting coordinates with classical baselines and NeuralEF. The results support PIEFS as a compact supervised spectral representation method and identify optimization stability, validation on explicit operator eigenproblems, and richer metric parameterizations as the main directions for future work.  \n1. Introduction  \nContributions.  \n• We formulate PIEFS as a supervised spectral-style representation method with sequential coordinate maps,  \nAccepted to the AI4Physics Workshop at ICML 2026 . 1Faculty of Computer Science, HSE University, Moscow, Russia 2AI VK, Moscow, Russia. Correspondence to: Varvara Nazarenko \u003C[varunaza@gmail.com](varunaza@gmail.com) >, Timur Lidzhiev \u003C[trlidzhiev@gmail.com](trlidzhiev@gmail.com) >, Alexander Tarakanov \u003C[atarakanov@hse.ru](atarakanov@hse.ru) > .  \nPreprint. July 8, 2026.  \nempirical Gram orthogonality, cross-entropy on a linear readout, and a modified Dirichlet penalty.  \n• We study three metric settings inside the Dirichlet term: identity (OFF), volume-preserving diagonal scaling (DIAG), and diagonal scaling after a structured Givens rotation (TROTTER ; apply order fixed as in Sec. 2.4) .  \n• We evaluate the same training pipeline across five benchmark settings (Table 2) against RF, LR, PCA+LR, and NeuralEF (Deng et al., 2022), with visualizationsof learned maps and training dynamics.  \nGraph-based spectral methods build a Laplacian on finite samples and use its eigenmaps as geometry-aware features (Gomez-Chova et al., 2008 ; Kunegis et al.) . Their theory connects graph eigenfunctions to Laplace–Beltrami modes when data concentrate on a manifold (Belkin & Niyogi, 2008), which explains strong performance in clustering and semi-supervised learning (Ng et al., 2001) . Two bottlenecks motivate mesh-free alternatives: eigencomputation cost grows with dataset size (Ford, 2015), and test-time evaluation at new points typically requires rebuilding the graph and its spectrum (Belkin & Niyogi, 2008) . Neural surrogates for operator eigenproblems are an active line of work (Jin et al., 2020 ; Deng et al., 2022 ; Choo et al., 2020) . Automatic differentiation makes it feasible to minimize Rayleigh-type objectives with PDE-style regularity while remaining discretization-free in the input domain (Feld et al., 2019 ; Lagaris et al., 1997) .  \nWe study physics-informed eigenfunction features with learnable scaling (PIEFS), implemented as learnable-metric Dirichlet coordinates: a sequence of scalar maps (ϕj) trained with a modified Dirichlet penalty that applies a datadependent linear map A (x) to gradients (Evans, 2022), together with batch Gram orthogonality and cross-entropy on linear logits. The outer schedule freezes earlier coordinates and updates one map at a time, yielding che","cbCaihh72j8aW4mx","https://ap.wps.com/l/cbCaihh72j8aW4mx","pdf",2031657,2,1,11,"English","en",105,"# Abstract\n# Introduction\n## Contributions\n# Methodology\n## Loss Function","[{\"question\":\"What is the role of the learnable metric A(x)=Λ(x)U(x) in PIEFS?\",\"answer\":\"A(x) transforms gradients inside the Dirichlet term: Λ(x) provides anisotropic (often diagonal) scaling, while U(x) is built from structured products of Givens rotations to model rotation-dependent scaling.\"}]",1784182781,28,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":78,"head_meta":80,"extra_data":82,"updated_unix":28},"physics-informed-eigenfunction-features-with-learnable-scaling-piefs","",{"@graph":36,"@context":77},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/physics-informed-eigenfunction-features-with-learnable-scaling-piefs/82765/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71],{"name":72,"@type":73,"acceptedAnswer":74},"What is the role of the learnable metric A(x)=Λ(x)U(x) in PIEFS?","Question",{"text":75,"@type":76},"A(x) transforms gradients inside the Dirichlet term: Λ(x) provides anisotropic (often diagonal) scaling, while U(x) is built from structured products of Givens rotations to model rotation-dependent scaling.","Answer","https://schema.org",{"og:url":51,"og:type":79,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":81,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":84},[85,89,93,97,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":86,"show_sort_weight":87,"slug":88},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":90,"show_sort_weight":91,"slug":92},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Exam",70,"exam",{"id":98,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},5,"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":98,"slug":130},19,"General","general"]