[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85009-en":3,"doc-seo-85009-105":29,"detail-sidebar-cat-0-en-105":91},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85009,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Higher-Order Geometric Updates for Levenberg–Marquardt Method via Riemann Normal Coordinates","Nonlinear least-squares optimization is a core computational task in regression, physics-informed neural networks, and machine learning. Parameters induce a manifold in data space, and parameter-effect curvature can dominate nonlinear behavior, exposing a limitation of the Levenberg–Marquardt (LM) method: its tangent-space direction is applied as a straight update in parameter coordinates. The paper proposes RNC-LM, using Riemann normal coordinates to reformulate geodesic-based acceleration into higher-order finite-step updates, paired with a line search. Experiments show improvements on curved valleys, numerical rank deficiency, PINN reaction–diffusion benchmarks, and large-scale potential fitting.","Higher-Order Geometric Updates for Levenberg–Marquardt Method via Riemann Normal Coordinates  \nJianing Liu 1, 2, ∗ and Dong H. Zhang2, 3, 4,†  \n1 State Key Laboratory of Chemical Reaction Dynamics and Department of Chemical Physics, University of Science and Technology of China, Hefei, 230026, China  \n2 State Key Laboratory of Chemical Reaction Dynamics,  \nDalian Institute of Chemical Physics, Chinese Academy of Sciences, Dalian, 116023, China.  \n3 University of Chinese Academy of Sciences, Beijing, 100049, China  \n4 Hefei National Laboratory, Hefei, 230088, China (Dated: July 9, 2026)  \narXiv :2607 .07623v 1 [ cs .LG] 8 Jul 2026  \nAbstract  \nNonlinear least-squares optimization is a central computational problem in regression, physicsinformed neural networks, and many other machine-learning tasks. Such problems have a natural geometric interpretation: as the parameters vary, the model predictions trace out a manifold in data space, and the parameter-effect curvature induced by a particular parameterization can become a dominant source of nonlinearity. This viewpoint reveals a fundamental limitation of the Levenberg–Marquardt (LM) method, whose tangent-space step is followed a straight update in parameter coordinates. Geodesic acceleration provides a second-order correction to LM, but it removes parameter-effect curvature only in the infinitesimal-step limit. Improving this geometric consistency for the finite steps used in actual optimization remains an open problem. Here we propose a Riemann-normal-coordinate Levenberg–Marquardt method, RNC-LM. By reformulating the geodesic equation, we extend second-order geodesic acceleration to arbitrary-order corrections, achieving finite-step updates with progressively higher reparameterization consistency. We further introduce a line search along the resulting RNC curve, which controls the finite distance traveled along a locally constructed geometric update while retaining a computational cost close to that of standard LM. We show that RNC coordinates modify the residual trajectory so that the tangential component of residual acceleration is eliminated order by order in the moving tangent frame, making the actual objective reduction more consistent with the linear prediction of LM. Experiments on classical nonlinear least-squares benchmarks demonstrate substantial improvements in problems involving curved valleys and numerical rank deficiency. On a challenging reaction–diffusion PINN benchmark, RNC-LM reduces the relative L2 error to the order of 10 −3 and recovers a physically meaningful solution under the original experimental setting. Finally, on a large-scale machine-learning potential-fitting task, RNC-LM achieves a 34× speedup over standard LM.  \n∗ Contact author: [liujn1218@mail.ustc.edu.cn](liujn1218@mail.ustc.edu.cn)  \n† Contact author: [zhangdh@dicp.ac.cn](zhangdh@dicp.ac.cn)  \nI. INTRODUCTION  \nModern machine learning and scientific computing increasingly rely on highly nonlinear and over-parameterized models, where the object of interest is usually not the parameter vector itself but the model it represents. Since different parameterizations can describe nearly the same model while inducing substantially different optimization behavior, the optimizer should ideally follow the geometry of the represented model rather than the arbitrary coordinates used to parametrize it.  \nThis concept has deeply influenced the development of modern optimizers through the use of appropriate local norms and metrics. While SGD defines steepest descent in Euclidean space [1], adaptive optimizers and matrix-preconditioned methods such as Adam and Shampoo can be interpreted as modifying the norm in which descent is measured [2, 3] . Natural gradient, Gauss–Newton (GN), and Levenberg–Marquardt (LM) methods make this structure explicit by using local metrics induced by KL divergence or residual maps, showing that optimizers differ not only in algorithmic mechanisms but also in the geometry used to","cbCaiq4JTaZOOt66","https://ap.wps.com/l/cbCaiq4JTaZOOt66","pdf",448132,1,34,"English","en",105,"# Abstract\n# I. Introduction","[{\"question\":\"What limitation of the Levenberg–Marquardt (LM) method motivates this work?\",\"answer\":\"LM computes an update direction from local tangent-space geometry, but applies it as a straight line in parameter coordinates, which becomes coordinate-dependent for finite steps and can conflict with the underlying geometric trajectory.\"},{\"question\":\"How does RNC-LM use Riemann normal coordinates to address finite-step geometric consistency?\",\"answer\":\"RNC-LM reformulates the geodesic equation to extend second-order geodesic acceleration into arbitrary-order corrections, producing finite-step updates with progressively higher reparameterization consistency.\"},{\"question\":\"What role does the line search play in RNC-LM?\",\"answer\":\"A line search is performed along the constructed RNC curve to control the finite distance traveled along a locally defined geometric update while keeping computational cost close to standard LM.\"}]",1784200207,86,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"higher-order-geometric-updates-for-levenbergmarquardt-method-via-riemann-normal-coordinates","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/higher-order-geometric-updates-for-levenbergmarquardt-method-via-riemann-normal-coordinates/85009/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What limitation of the Levenberg–Marquardt (LM) method motivates this work?","Question",{"text":75,"@type":76},"LM computes an update direction from local tangent-space geometry, but applies it as a straight line in parameter coordinates, which becomes coordinate-dependent for finite steps and can conflict with the underlying geometric trajectory.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does RNC-LM use Riemann normal coordinates to address finite-step geometric consistency?",{"text":80,"@type":76},"RNC-LM reformulates the geodesic equation to extend second-order geodesic acceleration into arbitrary-order corrections, producing finite-step updates with progressively higher reparameterization consistency.",{"name":82,"@type":73,"acceptedAnswer":83},"What role does the line search play in RNC-LM?",{"text":84,"@type":76},"A line search is performed along the constructed RNC curve to control the finite distance traveled along a locally defined geometric update while keeping computational cost close to standard LM.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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