[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81991-en":3,"doc-seo-81991-105":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},81991,687197207639,"Asher","https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd",8,"Research & Report","HPG-Diff: Hierarchical Physics-Guided Diffusion with Differentiable Connectivity Constraints for Topology Optimization","Deep generative models enable rapid design exploration for topology optimization, yet they lack intrinsic physics guidance, weakening generalization to unseen boundary conditions and producing floating material artifacts. HPG-Diff introduces Hierarchical Physics-Guided Diffusion with two complementary mechanisms. A hierarchical physics-guided strategy aligns precomputed physics features with the denoising process to target optimal load paths and improve generalizability. A floating material suppression loss, derived from a thermal conduction intuition, penalizes floating regions via differentiable connectivity constraints. Results report improved compliance errors and reduced floating ratios across in- and out-of-distribution tests, plus preliminary adaptation evidence.","arXiv :2607 .07233v 1 [ cs .LG] 8 Jul 2026  \nHPG-Diff: Hierarchical physics-guided diffusion with differentiable connectivity constraints for topology  \noptimization  \nJinbo Yanga,1 , Mingyue Yuanb,1 , Boyuan Zhangc , Yoshifumi Kitamurad ,  \nShikai Jinga,∗  \na School of Mechanical Engineering, Beijing Institute of  \nTechnology, Beijing, 100081, China  \nb Computer Science and Engineering, University of New South  \nWales, Sydney, 2052, Australia  \nc College of Intelligence and Computing, Tianjin University, Tianjin, 300072, China d Research Institute of Electrical Communication, Tohoku  \nUniversity, Sendai, 980-8576, Japan  \nAbstract  \nDeep generative models offer a promising paradigm for topology optimization, enabling rapid design exploration. However, these approaches lack intrinsic physics guidance, often leading to poor generalizability across unseen boundary conditions and the formation of floating material artifacts. To address these limitations, we propose Hierarchical Physics-Guided Diffusion (HPG-Diff), a novel diffusion framework that enforces physics consistency through two synergistic mechanisms. First, we introduce a hierarchical physics-guided strategy that aligns different precomputed physics features with the denoising process, guiding material distribution toward optimal load paths to enhance generalizability. Second, we propose a floating material suppression loss as a differentiable connectivity constraint inspired by thermal conduction to improve topological connectivity. By simulating a virtual heat propagation process from load positions, this mechanism explicitly penalizes floating material during training. Quantitative evaluations demonstrate that HPG-Diff achieves average compliance errors of 0.87%(in-distribution) and 5.29%(out-of-distribution), while reducing floating material ratios to  \n∗ Corresponding author. E-mail [address: jingshikai@bit.edu.cn](address: jingshikai@bit.edu.cn) (S. Jing)  \n1 These authors contributed equally to this work.  \n2.90% and 2 .44%, respectively. Furthermore, case studies on a 3:1 rectangular domain, including cantilever and bridge benchmarks, provide preliminary evidence that lightweight LoRA fine-tuning with a small dataset can support the adaptation of HPG-Diff to rectangular non-square domains.  \nKeywords: Topology optimization, Diffusion models, Physics-guided generation, Connectivity constraints, Generative design  \n1. Introduction  \nTopology optimization (TO) has been widely used in various engineering design scenarios due to its capacity to generate lightweight structures with high mechanical performance. To date, numerous TO designed structures have been successfully applied in the aerospace [1, 2, 3], mechanical [4, 5, 6], civil [7, 8], and biomedical [9, 10, 11] fields. This iterative optimization approach typically relies on Finite Element Analysis (FEA) . A common example is the Solid Isotropic Material with Penalization (SIMP) [12] . However, it is computationally demanding, especially in large-scale optimization problems.  \nData-driven design methods have also been widely explored in broader engineering domains [13, 14, 15, 16, 17] . In topology optimization, deep learning has significantly reduced computational costs using surrogate models [18, 19, 20, 21, 22] . However, they often suffer from a lack of physics consistency: the mechanical performance and manufacturability may violate the physics principles. Since surrogate models approximate complex behavior by fitting input–output relationships, they may not effectively capture the physics representation of optimization objectives and constraints, potentially resulting in generated structures with anomalous stress concentrations or connectivity defects. For example, density-field surrogate predictions may require additional physics-based correction or post-processing when the learned mapping does not fully preserve equilibrium, volume, or connectivity-related constraints [23, 24] . At the same time, prac","cbCaigu9hBQ7aUQO","https://ap.wps.com/l/cbCaigu9hBQ7aUQO","pdf",1958123,5,1,43,"English","en",105,"# Introduction\n## Background and limitations of existing topology optimization approaches\n## Motivation: efficiency–quality trade-off and need for fast diverse generation","[{\"question\":\"What problem does HPG-Diff address in diffusion-based topology optimization?\",\"answer\":\"It addresses poor physics guidance that can lead to weak generalization across unseen boundary conditions and floating material artifacts in generated designs.\"},{\"question\":\"How does HPG-Diff improve generalizability during generation?\",\"answer\":\"It uses a hierarchical physics-guided strategy that aligns different precomputed physics features with the denoising process, steering material distributions toward optimal load paths.\"},{\"question\":\"What is the role of the floating material suppression loss?\",\"answer\":\"It acts as a differentiable connectivity constraint inspired by thermal conduction, simulating virtual heat propagation from load positions to explicitly penalize floating material during 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problem does HPG-Diff address in diffusion-based topology optimization?","Question",{"text":76,"@type":77},"It addresses poor physics guidance that can lead to weak generalization across unseen boundary conditions and floating material artifacts in generated designs.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does HPG-Diff improve generalizability during generation?",{"text":81,"@type":77},"It uses a hierarchical physics-guided strategy that aligns different precomputed physics features with the denoising process, steering material distributions toward optimal load paths.",{"name":83,"@type":74,"acceptedAnswer":84},"What is the role of the floating material suppression loss?",{"text":85,"@type":77},"It acts as a differentiable connectivity constraint inspired by thermal conduction, simulating virtual heat propagation from load positions to explicitly penalize floating material during 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