[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83479-en":3,"doc-seo-83479-105":30,"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":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},83479,1099513958762,"Logic","https://ap-avatar.wpscdn.com/avatar/1000023916a998db790?x-image-process=image/resize,m_fixed,w_180,h_180&k=1784791008015729253",8,"Research & Report","WarpagePINN: Thermal Warpage Prediction in Advanced Packaging via a Two-Stage Physics-Informed Neural Networks","Thermal warpage is a reliability bottleneck in advanced packaging, driven mainly by mismatched thermal expansion coefficients (CTE) across heterogeneously integrated materials. The work presents a two-stage physics-informed neural network framework, WarpagePINN, for chiplet-level coupled thermal–warpage prediction. Temperature and deformation are learned without labeled simulator data by enforcing governing equations in the training loss. A Fourier-series temperature model satisfies boundary conditions, while an MLP predicts warpage using an energy-based hybrid supervisory strategy. A parametric variant quantifies CTE uncertainty, matching finite elements with MAE 0.2 μm and achieving about 1000× speedup in CTE studies.","WarpagePINN: Thermal Warpage Prediction in Advanced Packaging via a Two-Stage Physics-Informed Neural Networks  \nXinyu Li, Min Tang, Senior Member, IEEE, Zeyu Sun, Wenxing Zhu, Jianhua Zhang, and  \nLiang Chen, Member, IEEE  \narXiv :2607 .00364v1 [math .NA] 1 Jul 2026  \nAbstract—Thermal warpage has become a critical issue in advanced packaging, primarily caused by the mismatch in coefficients of thermal expansion (CTE) among heterogeneously integrated materials. However, only a limited number of studies have focused on developing computational methods for coupled thermal-warpage prediction in the chiplet. This paper proposesa two-stage physics-informed neural network (WarpagePINN) framework to compute both temperature profile and warpage deformation of chiplets. The neural networks are trained without relying on labeled datasets generated by conventional simulators. In the first stage, the temperature field is modeled using a Fourier series representation that inherently satisfies boundary conditions, and the network is trained solely through a loss function derived from the governing equation. In the second stage, a multilayer perceptron (MLP) is employed for warpage prediction, utilizing a novel hybrid supervisory strategy to optimize the energy-based loss function instead of residual loss. A parametric WarpagePINN is also developed to quantify uncertainties associated with the CTE. Numerical results show that the proposed WarpagePINN framework achieves excellent agreement with conventional finite element methods, with a mean absolute error (MAE) of 0.2 µm, while achieving a speedup of approximately 1000 × in CTE parameterization studies.  \nIndex Terms—Advanced packaging, warpage, temperature, CTE, physics-informed neural networks, a hybrid supervisory strategy.  \nI. Introduction  \nA  \nDVANCED packaging technology for heterogeneous integration has become  \nmulti-chiplet a promising  \nalternative to transistor-level miniaturization for increasing integrated circuit density, effectively addressing the growing demands of high-performance computing (HPC)  \nThis work was supported in part by National Natural Science Foundation of China under Grant 92473105 and 62504151; in part by National Key Research and Development Program of China under Grant 2025YFA1213000; and in part by State Key Laboratory of Radio Frequency Heterogeneous Integration (Open Scientific Research Program No. KF2024005) . (Corresponding author: Liang Chen.)  \nX. Li, J. Zhang, and L. Chen are with the School of Microelectronics and Shanghai Key Laboratory of Chips and Systems for Intelligent Connected Vehicle, Shanghai University, Shanghai 201800, China (email: [lchenshu@shu.edu.cn](lchenshu@shu.edu.cn)).  \nM. Tang is with the State Key Laboratory of Radio Frequency Heterogeneous Integration, Shanghai Jiao Tong University, Shanghai, 200240, China.  \nZ. Sun is with the Institute of Microelectronics, Chinese Academy of Sciences, Beijing, 100029, China.  \nW. Zhu is with the Center for Discrete Mathematics and Theoretical Computer Science, Fuzhou University, Fuzhou 350108, China.  \napplications, including artificial intelligence (AI), large language models (LLMs) and autonomous vehicles [1] . However, the use of disparate materials in such packages introduces a mismatch in their coefficients of thermal expansion (CTE) [2], [3] . Moreover, continual increases in packaging dimensions, exemplified by state-of-the-art TSMC CoWoS technology with chip-on-wafer (CoW) sizes reaching 70 × 70 mm2 and substrates up to 100 × 100 mm2 , have made thermal warpage a critical reliability issue [4], as shown in Fig. 1. Therefore, accurate prediction and mitigation of thermal warpage at early design stages are essential for modern heterogeneous integration systems [5] .  \n(a)  \n(b)  \nFig. 1. (a) Cross-sectional view of the TSMC CoWoS-R package. (b) Thermomechanical warpage of the package induced by heterogeneous integration and thermal expansion.  \nNumerous modeling approaches have been","cbCainddd4FWunZD","https://ap.wps.com/l/cbCainddd4FWunZD","pdf",8994281,4,1,10,"English","en",105,"# Introduction\n## Thermal warpage background and challenges\n## Modeling approaches: analytical and FEM\n## Physics-informed neural networks and PINNs\n## Proposed WarpagePINN framework","[{\"question\":\"What causes thermal warpage in advanced packaging according to the document?\",\"answer\":\"Thermal warpage primarily results from mismatches in coefficients of thermal expansion (CTE) among heterogeneously integrated materials, which lead to coupled thermal and mechanical deformation.\"},{\"question\":\"How does WarpagePINN predict temperature and warpage without labeled datasets?\",\"answer\":\"In the first stage, the temperature field is represented with a Fourier series that satisfies boundary conditions and the network is trained using a loss derived from the governing equation. In the second stage, an MLP predicts warpage using a hybrid supervisory strategy that optimizes an energy-based loss.\"},{\"question\":\"What performance does the framework achieve compared with finite element methods?\",\"answer\":\"Numerical results show excellent agreement with conventional finite element methods, with a mean absolute error (MAE) of 0.2 μm and an approximately 1000× speedup in CTE parameterization studies.\"}]",1784188301,25,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"warpagepinn-thermal-warpage-prediction-in-advanced-packaging-via-a-two-stage-physics-informed-neural-networks","",{"@graph":36,"@context":85},[37,53,68],{"@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":20},"https://docshare.wps.com/document/warpagepinn-thermal-warpage-prediction-in-advanced-packaging-via-a-two-stage-physics-informed-neural-networks/83479/",{"url":52,"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",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What causes thermal warpage in advanced packaging according to the document?","Question",{"text":75,"@type":76},"Thermal warpage primarily results from mismatches in coefficients of thermal expansion (CTE) among heterogeneously integrated materials, which lead to coupled thermal and mechanical deformation.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does WarpagePINN predict temperature and warpage without labeled datasets?",{"text":80,"@type":76},"In the first stage, the temperature field is represented with a Fourier series that satisfies boundary conditions and the network is trained using a loss derived from the governing equation. In the second stage, an MLP predicts warpage using a hybrid supervisory strategy that optimizes an energy-based loss.",{"name":82,"@type":73,"acceptedAnswer":83},"What performance does the framework achieve compared with finite element methods?",{"text":84,"@type":76},"Numerical results show excellent agreement with conventional finite element methods, with a mean absolute error (MAE) of 0.2 μm and an approximately 1000× speedup in CTE parameterization studies.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":22,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":22,"slug":133},"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]