[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86132-en":3,"doc-seo-86132-105":30,"detail-sidebar-cat-0-en-105":92},{"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},86132,962075114765,"Quinn","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","Multi-dimensional Training Priority Weighting Based on Physical Information Propagation Paths","Physics-informed neural networks (PINNs) solve partial differential equations (PDEs) but often optimize region residuals and constraints with equal weight in parallel, harming stability and accuracy when the solution should form progressively “from source to response.” A unified training-priority concept is introduced by mapping priorities to the PDE’s physical information propagation paths, covering temporal, spatial, and boundary cases. Neural tangent kernel analysis explains why standard PINNs lack a definite priority order, then a residual-weighting framework converts propagation order into loss-level priority. Multiple benchmarks validate improved convergence and prediction without changing architecture.","MULTI-DIMENSIONAL TRAINING-PRIORITY WEIGHTING BASED ON PHYSICAL INFORMATION PROPAGATION PATHS: A UNIFIED RESIDUAL-WEIGHTING FRAMEWORK FOR PHYSICS-INFORMED  \nNEURAL NETWORKS  \narXiv :2607 . 1 1094v 1 [ cs .LG] 13 Jul 2026  \nZhangyi Lian  \nCollege of Mechanical and Vehicle Engineering Chongqing University Chongqing 400030, China  \nXinda Dong  \nState Key Laboratory of Tribology in Advanced Equipment Department of Mechanical Engineering Tsinghua University  \nBeijing 100084, China  \nWenxuan Huo  \nState Key Laboratory of Tribology in Advanced Equipment  \nDepartment of Mechanical Engineering  \nTsinghua University  \nBeijing 100084, China  \nWeifeng Huang  \nState Key Laboratory of Tribology in Advanced Equipment  \nDepartment of Mechanical Engineering  \nTsinghua University  \nBeijing 100084, China  \nGang Zhu  \nState Key Laboratory of Tribology in Advanced Equipment  \nDepartment of Mechanical Engineering  \nTsinghua University  \nBeijing 100084, China  \nQiang He∗  \nState Key Laboratory of Tribology in Advanced Equipment  \nDepartment of Mechanical Engineering  \nTsinghua University  \nBeijing 100084, China  \n[heqiang@tsinghua.edu.cn](heqiang@tsinghua.edu.cn)  \nABSTRACT  \nPhysics-informed neural networks (PINNs) have shown considerable promise for solving partial differential equations (PDEs); however, their synchronous optimization mechanism treats the residuals of different regions and different constraints with equal weight and in parallel, which is often inconsistent with the process by which the solution forms progressively “from source to response”along the physical information propagation path, thereby degrading training stability and solution accuracy. Existing causal training methods focus mainly on the temporal dimension and lack a unified characterization of the spatial and boundary dimensions. To address this, we define a unified class of training priorities according to the propagation path of physical information: along the  \nPDE’s physical information propagation path, premise regions should be learned before the regions that depend on them; the temporal, spatial, and boundary priorities are precisely instances of this principle on different propagation paths. On this basis, using neural tangent kernel (NTK) training dynamics, we theoretically analyze why the standard PINN does not obey this priority: its residual convergence order is governed by the NTK spectrum and is independent of the propagation path, and the spectral bias does not actively favor premise regions, so it does not establish a definite convergence order. Accordingly, we propose a unified multi-dimensional priority-constraint framework that, by partitioning the domain in order along the propagation path and constructing negative-exponential residual weights, converts the physical propagation order into a training priority at the loss level.  \nFor cases in which multiple priorities coexist, we introduce a directional compatibility coefficient to clarify the applicability boundary that “orthogonal directions can be coupled multiplicativelyin synergy, whereas coaxial opposite directions cannot.” Multiple benchmark cases show that the proposed method consistently improves the convergence behavior and prediction accuracy of PINNson problems with a clear propagation path or a constraint-dominated structure, without modifying the network architecture and with controllable additional computational cost.  \nKeywords physics-informed neural networks · training priority · physical information propagation path · residual weighting · neural tangent kernel · multi-dimensional coupling  \n1 Introduction  \nPhysics-informed neural networks (PINNs), an emerging scientific-computing paradigm [1–4], have shown broad application prospects in solving partial differential equations (PDEs) in recent years [5–9] . The core advantage of PINNs lies in their ability to embed the PDE, the initial condition, and the boundary conditions uniformly into a single loss function, thereby avoiding the ","cbCaibcp72bZMdNX","https://ap.wps.com/l/cbCaibcp72bZMdNX","pdf",1676211,5,1,23,"English","en",105,"# Abstract\n# Introduction\n## PINNs and synchronized optimization assumptions\n## Prior work: architecture, loss balancing, decomposition, adaptive weighting\n## Physical-information propagation and causal training\n## Need for unified spatial, temporal, and boundary priorities","[{\"question\":\"Why do standard PINNs sometimes have degraded training stability and solution accuracy?\",\"answer\":\"They optimize residuals from different regions and constraints with equal weight and in parallel, which may conflict with how the PDE solution forms along the physical information propagation path.\"},{\"question\":\"How does the proposed framework define training priorities for PINNs?\",\"answer\":\"It defines a unified class of training priorities based on physical information propagation paths, requiring premise regions to be learned before dependent regions, and treating temporal, spatial, and boundary priorities as concrete instances.\"},{\"question\":\"What theoretical and practical mechanism is used to implement the priority in training?\",\"answer\":\"Neural tangent kernel (NTK) training dynamics are used to analyze the absence of priority in standard PINNs, and the method partitions the domain along the propagation path and applies negative-exponential residual weights so propagation order becomes a loss-level training priority.\"}]",1784208773,58,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"multi-dimensional-training-priority-weighting-based-on-physical-information-propagation-paths","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/multi-dimensional-training-priority-weighting-based-on-physical-information-propagation-paths/86132/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why do standard PINNs sometimes have degraded training stability and solution accuracy?","Question",{"text":76,"@type":77},"They optimize residuals from different regions and constraints with equal weight and in parallel, which may conflict with how the PDE solution forms along the physical information propagation path.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the proposed framework define training priorities for PINNs?",{"text":81,"@type":77},"It defines a unified class of training priorities based on physical information propagation paths, requiring premise regions to be learned before dependent regions, and treating temporal, spatial, and boundary priorities as concrete instances.",{"name":83,"@type":74,"acceptedAnswer":84},"What theoretical and practical mechanism is used to implement the priority in training?",{"text":85,"@type":77},"Neural tangent kernel (NTK) training dynamics are used to analyze the absence of priority in standard PINNs, and the method partitions the domain along the propagation path and applies negative-exponential residual weights so propagation order becomes a loss-level training priority.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & 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