[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-119982-en":3,"doc-seo-119982-105":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":4,"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":27,"seo_description":14,"update_tm":28,"read_time":29},119982,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","MITIGATING DISTRIBUTION SHIFT IN MACHINE LEARNING-AUGMENTED HYBRID SIMULATION","The work investigates distribution shift that arises in machine-learning augmented hybrid simulation, where parts of simulation algorithms are replaced by data-driven surrogates. A mathematical framework is developed to characterize the structure of these problems and to explain the mechanisms driving shift. The study links distribution shift to simulation error through both numerical results and theoretical arguments, then introduces a tangent-space regularized estimator to control the shift and improve long-term accuracy. In linear dynamics, the method’s effectiveness is analyzed rigorously and verified in experiments on reaction-diffusion and Navier-Stokes systems, yielding substantial gains especially under strong shift conditions.","arXiv :2401 .09259v2 [math .NA] 15 Jun 2025  \nMITIGATING DISTRIBUTION SHIFT IN MACHINE LEARNING-AUGMENTED HYBRID SIMULATION  \nJIAXI ZHAO∗ AND QIANXIAO LI†  \nAbstract.  \nWe study the problem of distribution shift generally arising in machine-learning augmented hybrid simulation, where parts of simulation algorithms are replaced by data-driven surrogates. A mathematical framework is established to understand the structure of machine-learning augmented hybrid simulation problems and the cause and effect of the associated distribution shift. We show correlations between distribution shift and simulation error both numerically and theoretically. Then, we propose a simple methodology based on tangent-space regularized estimator to control the distribution shift, thereby improving the long-term accuracy of the simulation results. In the linear dynamics case, we provide a thorough theoretical analysis to quantify the effectiveness of the proposed method. Moreover, we conduct several numerical experiments, including simulating a partially known reaction-diffusion equation and solving Navier-Stokes equations using the projection method with a data-driven pressure solver. In all cases, we observe marked improvements in simulation accuracy under the proposed method, especially for systems with high degrees of distribution shift, such as those with relatively strong non-linear reaction mechanisms, or flows at large Reynolds numbers.  \nKey words. machine learning, distribution shift, regularization, error analysis, fluid dynamics MSC codes. 68T99, 65M15, 37M05  \n1. Introduction. Many scientific computational applications, such as computational fluid dynamics (CFD) and molecular dynamics (MD) can be viewed as dynamical system modeling and simulation problems, which are tackled by rigorous numerical tools with theoretical guarantee [53] . However, in many cases a part of the simulation workflow, such as the Reynolds stresses in Reynolds-averaged Navier-Stokes equation (RANS) [1] and the exchange-correlation energies in density function theory used to compute force fields that drive MD simulations [30], depends on models that are either expensive to compute or even unknown in practice. One thus often resorts toa hybrid simulation method, where the known, resolved components of the dynamics are computed exactly, while the unresolved components are replaced by approximate, but computationally tractable models. For example, solving Navier-Stokes equations using the projection method involves two steps. In the first step, all the terms except for the gradient of the pressure are used to evolve the velocity. This step is computationally cheap and thus understood as the resolved part. Next, the pressure is solved from a Poisson equation and then used to correct the velocity. Most of the computational cost is contained in solving this Poisson equation, and we thereby viewed this as the unresolved part. We call such scientific computing problems with both resolved and unresolved parts “hybrid simulation problems”. Similar problems are surveyed in [59] under the name of “Hybrid physics-DL models”.  \nAs machine learning becomes increasingly powerful in areas like computer vision and natural language processing, practitioners begin to use data-driven modules to model the unresolved part to carry out the simulation. For example, in [57] the author replaced the numerical Poisson solver of the unresolved part with a convolutional neural network trained using a novel unsupervised learning framework. Then, this data-driven model is coupled with resolved models and provides fast and realistic  \n∗ Department of Mathematics, National University of Singapore, 117543, Singapore (ji[axi.zhao@u.nus.edu](axi.zhao@u.nus.edu)).  \n†Department of Mathematics & Institute for Functional Intelligent Materials, National University of Singapore, 117543, Singapore ([qianxiao@nus.edu.sg](qianxiao@nus.edu.sg)).  \n2 ZHAO AND LI  \nsimulation results in 2D and 3D. Similar ideas are","cbCailC0dXsa8tvi","https://ap.wps.com/l/cbCailC0dXsa8tvi","pdf",2832433,1,33,"English","en",105,"# Introduction\n## Hybrid simulation problems and ML augmentation\n## Distribution shift and existing mitigation approaches\n## Core methodology: tangent-space regularized estimator\n## Theoretical analysis in linear dynamics\n## Numerical experiments and observed improvements","[{\"question\":\"What causes distribution shift in machine-learning augmented hybrid simulation?\",\"answer\":\"Distribution shift occurs when iteratively applying a data-driven surrogate in the simulation drives the system into regimes not represented by the training distribution.\"},{\"question\":\"How does the paper connect distribution shift with simulation error?\",\"answer\":\"It establishes correlations between distribution shift and simulation error using both numerical experiments and theoretical analysis.\"},{\"question\":\"What method is proposed to mitigate distribution shift?\",\"answer\":\"The paper proposes a tangent-space regularized estimator to control distribution shift, thereby improving long-term simulation accuracy.\"}]","MITIGATING DISTRIBUTION SHIFT IN MACHINE LEARNING-AUGMENTED HYBRID SIMULATION | PDF",1785727465,83,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"mitigating-distribution-shift-in-machine-learning-augmented-hybrid-simulation","",{"@graph":36,"@context":85},[37,54,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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/mitigating-distribution-shift-in-machine-learning-augmented-hybrid-simulation/119982/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":62,"encodingFormat":61,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-03",true,{"@type":65,"interactionType":66,"userInteractionCount":4},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What causes distribution shift in machine-learning augmented hybrid simulation?","Question",{"text":75,"@type":76},"Distribution shift occurs when iteratively applying a data-driven surrogate in the simulation drives the system into regimes not represented by the training distribution.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the paper connect distribution shift with simulation error?",{"text":80,"@type":76},"It establishes correlations between distribution shift and simulation error using both numerical experiments and theoretical analysis.",{"name":82,"@type":73,"acceptedAnswer":83},"What method is proposed to mitigate distribution shift?",{"text":84,"@type":76},"The paper proposes a tangent-space regularized estimator to control distribution shift, thereby improving long-term simulation accuracy.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":89,"canonical":52},"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":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":53,"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":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]