[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81745-en":3,"doc-seo-81745-105":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},81745,4398048949847,"Eliana","https://ap-avatar.wpscdn.com/avatar/400002536579ef2da7f?_k=1778318612642679267",8,"Research & Report","Sparse Nonlinear Accelerated Projection for Physics-Constrained Generative Modeling","Physics-constrained generative models act as scalable surrogates for physical simulation, but their outputs can violate conservation laws, boundary conditions, and nonlinear invariants. Constrained sampling enforces these constraints at inference time, yet nonlinear projection requires repeated correction and trajectory optimization, making it computationally expensive, especially under limited sparse solver support in ML frameworks. The approach accelerates constraint projection by exploiting block-sparse Jacobian and KKT structure from local PDE couplings, using ExaModels.jl, MadNLP.jl, and GPU sparse factorization. Applied to Physics-Constrained Flow Matching across PDE benchmarks, it preserves constraint satisfaction while speeding nonlinear optimization.","SNAP-FM: Sparse Nonlinear Accelerated Projection for Physics-Constrained  \nGenerative Modeling  \nAlaina Kolli * 1 Theodoros Xenakis * 1 2 Utkarsh Utkarsh * 1 Pengfei Cai 1 Rafael Gmez-Bombarelli 1  \nAlan Edelman 1 Christopher V. Rackauckas 1  \narXiv :2607 .00095v1 [ cs .LG] 30 Jun 2026  \nAbstract  \nGenerative models have emerged as scalable surrogates for physical simulation, yet they offer no guarantee that their outputs respect the conservation laws, boundary conditions, and nonlinear invariants that govern the underlying physics. Constrained sampling closes this gap, enforcing such constraints exactly at inference time without retraining, but at a computational cost: projection, correction and trajectory-optimization steps are repeated during sampling, with these steps becoming expensive for nonlinear constraints. Standard ML frameworks exacerbate this: their dense tensor algebra and limited sparse solver composability obscure the structure that physical constraints naturally induce, making efficient batched nonlinear optimization difficult to realize in practice. We address this bottleneck by exploiting the structure that sample-wise batching and local PDEcouplings induce in the projection subproblems – namely, block-sparse Jacobian and KKT systems – exposing this structure using ExaModels . jland solving the resulting sparse nonlinear programs with MadNLP . jl and GPU sparse factorization. Applied to Physics-Constrained Flow Matching (PCFM), on PDE benchmarks with linear, nonlinear, one-dimensional, and twodimensional constraints, this approach accelerates nonlinear constraint projection while maintaining constraint satisfaction. These results show that sparse GPU nonlinear optimization is a practical foundation for constrained generative sampling in scientific machine learning.  \n1Massachusetts Institute of Technology 2Norwegian University of Science and Technology. Correspondence to: Christopher V. Rackauckas \u003C[crackauc@mit.edu](crackauc@mit.edu) > .  \nProceedings of the AI4Physics Workshop at the 43 rd International Conference on Machine Learning (AI4Physics@ICML 2026) , Seoul, South Korea. 2026. Copyright 2026 by the author(s) .  \n1. Introduction  \nGenerative models have emerged as flexible surrogates for physical simulation, learning solution distributions for partial differential equations (PDEs) and amortizing inference across varying physical conditions (Price et al., 2023 ; Yuan et al., 2023 ; Huang et al., 2024 ; Utkarsh et al., 2025a) . Yet their deployment in scientific settings is limited by a fundamental gap: unconstrained generative models do not, by themselves, guarantee physical fidelity. Conservation of mass, momentum, and energy, nonlinear boundary conditions, and invariants tied to the governing equations are central to classical numerical simulation, but are routinely violated by learned surrogates unless explicitly enforced (Raissi et al., 2019 ; Li et al., 2021) . Closing this gap without sacrificing the amortized inference advantage of generative models is the central challenge of physics-constrained generative modeling.  \nConstraint enforcement in generative sampling can be broadly divided into soft and hard approaches. Soft methods, including training-time penalty losses (Baldan et al., 2025 ; Huang et al., 2024), PINN-style residual regularization, physics-informed neural operators, and architecturelevel inductive biases (Greydanus et al., 2019 ; RichterPowell et al., 2022), encourage constraint satisfaction approximately. These approaches are computationally attractive and scale naturally within modern deep learning pipelines, but they generally provide no exact feasibility guarantees and can exhibit increased constraint violation under distribution shift.  \nHard-constrained methods instead aim to enforce feasibility exactly, either through inference-time correction and optimization (Utkarsh et al., 2025a ; Christopher et al., 2024 ; Cheng et al., 2025 ; Rmer et al., 2024 ; Yuan et al., 2023) o","cbCaikT7K6zSMAr1","https://ap.wps.com/l/cbCaikT7K6zSMAr1","pdf",847935,4,1,17,"English","en",105,"# Introduction\n## Gap in unconstrained generative fidelity\n## Soft vs hard constraint enforcement\n## Projection-based computational bottleneck","[{\"question\":\"Why do unconstrained generative models fail in physics-based applications?\",\"answer\":\"They do not guarantee physical fidelity, so learned surrogates can violate conservation laws, nonlinear boundary conditions, and equation-linked invariants that classical solvers respect.\"},{\"question\":\"What distinguishes soft constraint methods from hard-constrained sampling?\",\"answer\":\"Soft methods encourage constraints approximately via training-time penalties or residual regularization, while hard methods enforce feasibility exactly at inference time through correction/optimization or end-to-end constrained formulations.\"},{\"question\":\"How does the proposed method speed up nonlinear constraint projection?\",\"answer\":\"It exploits constraint-induced sparsity by leveraging block-sparse Jacobian and KKT system structure in projection subproblems, then solves the resulting sparse nonlinear programs using ExaModels.jl, MadNLP.jl, and GPU sparse factorization.\"}]",1784175796,43,{"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},"sparse-nonlinear-accelerated-projection-for-physics-constrained-generative-modeling","",{"@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/sparse-nonlinear-accelerated-projection-for-physics-constrained-generative-modeling/81745/",{"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","2026-07-25","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},"Why do unconstrained generative models fail in physics-based applications?","Question",{"text":75,"@type":76},"They do not guarantee physical fidelity, so learned surrogates can violate conservation laws, nonlinear boundary conditions, and equation-linked invariants that classical solvers respect.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What distinguishes soft constraint methods from hard-constrained sampling?",{"text":80,"@type":76},"Soft methods encourage constraints approximately via training-time penalties or residual regularization, while hard methods enforce feasibility exactly at inference time through correction/optimization or end-to-end constrained formulations.",{"name":82,"@type":73,"acceptedAnswer":83},"How does the proposed method speed up nonlinear constraint projection?",{"text":84,"@type":76},"It exploits constraint-induced sparsity by leveraging block-sparse Jacobian and KKT system structure in projection subproblems, then solves the resulting sparse nonlinear programs using ExaModels.jl, MadNLP.jl, and GPU sparse factorization.","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},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & 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