[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81860-en":3,"doc-seo-81860-105":29,"detail-sidebar-cat-0-en-105":90},{"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":13,"seo_description":14,"update_tm":27,"read_time":28},81860,5909877438554,"Maeve","https://ap-avatar.wpscdn.com/avatar/5600025385ad2bf12a7?_k=1778553567797529272",8,"Research & Report","In-span Learning: Adapting Reduced-Order Models Using Their Own Predictions","Reduced-order models (ROMs) accelerate simulation by compressing high-dimensional dynamics into low-dimensional subspaces, but they lose accuracy when online states drift beyond the training regime. Adaptive ROMs typically update subspaces using external out-of-span information such as full-order corrections or sensor snapshots. This work identifies an additional, previously underused in-span adaptation channel within the existing reduced subspace. Streaming the model’s own predictions through incremental SVD with forgetting yields a trajectory-informed spectral preconditioner that reweights and realigns the basis without changing the subspace, improving absorption of future out-of-span corrections. The mechanism is analyzed in a 3D spiral and validated on viscous Burgers and Fisher–KPP dynamics, linking in-span learning to in-context learning and offering guidance for computational science.","arXiv :2607 .02937v 1 [ cs .LG] 3 Jul 2026  \n\n| In-span learning: adapting reduced-order models using their own predictions |\n| --- |\n| Amirpasha Hedayat1,3,‗ Laura Balzano2,3 Karthik Duraisamy1,3\u003Cbr>1 Department of Aerospace Engineering\u003Cbr>2 Department of Electrical Engineering and Computer Science\u003Cbr>3 Michigan Institute for Computational Discovery and Engineering, University of Michigan, Ann Arbor, MI, USA\u003Cbr>Abstract\u003Cbr>Reduced-order models compress high-dimensional dynamics into low-dimensional representations that can be evaluated rapidly, but they lose accuracy when online dynamics drift beyond the training data. Adaptive methods address this by updating the subspace online with external, out-of-span information, such as full-order corrections or sensor snapshots. We discovered that a complementary and previously unexploited in-span adaptation channel exists within the current reduced subspace.\u003Cbr>By streaming the model’s own predictions through an incremental singular-value decomposition with forgetting, we obtain a trajectory-informed spectral preconditioner, in which the subspace is unchanged but the basis is reweighted and realigned toward the modes visited by the dynamics. This enables the model to absorb future out-of-span corrections more effectively. We expose aspects of this mechanism on a three-dimensional spiral and confirm it on viscous Burgers and Fisher–KPP dynamics. We also discuss how in-span learning can be viewed as a dynamicalsystems analogue of in-context learning. More broadly, in-span learning suggests anew principle for computational science, revealing that model-generated trajectories contain more usable information than previously recognized.\u003Cbr>Keywords: In-span learning, Spectral preconditioning, Solver acceleration, Adaptive model reduction, Incremental SVD, Subspace tracking. |\n\nThis is a preprint of a manuscript submitted to Nature Computational Science. The Supplementary Information of the submitted version is included here as Appendices A–E.  \n1 Introduction  \nAcross science and engineering, applications such as aircraft design [1], weather forecasting [2, 3, 4], combustion predictions [5], power-grid control [6], uncertainty quantification [7, 8], and digital twins [9, 10] all rely on numerical models that require expensive computations. There is also a need to run these computations repeatedly across parameters, controls, uncertain inputs,  \n‗Corresponding author: [ahedayat@umich.edu](ahedayat@umich.edu)  \nHedayat et al. 1  \nIn-span learning: adapting reduced-order models using their own predictions  \nor real-time decisions. This setting calls for simulators that are accurate enough to trust and inexpensive enough to use freely.  \nReduced-order models (ROMs) address this need by compressing high-dimensional simulations into low-dimensional representations [11, 12, 13, 14] . Although a discretized physical model may contain millions of degrees of freedom, its relevant trajectories often occupy a much smaller region of state space. Projection-based ROMs exploit this structure by constructing a low-dimensional subspace, commonly from proper orthogonal decomposition (POD) of full-order snapshots [15, 16, 17], and evolving only the coordinates of the solution in that subspace [18] . When the online dynamics remain close to the training data, this compression can deliver orders-of-magnitude speed-ups [21] . The same compression also creates the central failure mode of ROMs. A static ROM is trained offline and deployed with a fixed basis. It can interpolate accurately within the regime represented by its snapshots, but it loses predictive power when the online trajectory drifts beyond that regime. This is a form of distribution shift that exists whether the reduced representation is a linear subspace, a nonlinear manifold [23], or a learned surrogate [22] . For projection-based ROMs considered in this work, this limitation is especially severe for systems with slowly decaying Kolmogorov 􀀣-widt","cbCaifujBjL7kO9J","https://ap.wps.com/l/cbCaifujBjL7kO9J","pdf",3036062,1,32,"English","en",105,"# Introduction\n## Reduced-order models and distribution shift\n## Adaptive ROMs and out-of-span updates\n## In-span adaptation channel and incremental spectral update","[{\"question\":\"Why do reduced-order models often fail when simulations move away from training data?\",\"answer\":\"A static ROM uses a fixed basis trained offline, so it interpolates well only within the snapshot regime. When the online trajectory drifts, this distribution shift reduces predictive power.\"},{\"question\":\"What is the standard adaptive approach for ROMs?\",\"answer\":\"Adaptive ROMs evolve the reduced representation online using out-of-span information with nonzero residual outside the current subspace, such as full-order corrections or sensor snapshots.\"},{\"question\":\"How does the proposed in-span learning mechanism differ from out-of-span adaptation?\",\"answer\":\"In-span learning streams the ROM’s own predictions, which remain inside the current reduced subspace, to reorganize the representation through incremental SVD with forgetting. The subspace is unchanged, but the basis is reweighted and realigned toward visited modes.\"}]",1784176704,81,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"in-span-learning-adapting-reduced-order-models-using-their-own-predictions","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/in-span-learning-adapting-reduced-order-models-using-their-own-predictions/81860/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"Why do reduced-order models often fail when simulations move away from training data?","Question",{"text":74,"@type":75},"A static ROM uses a fixed basis trained offline, so it interpolates well only within the snapshot regime. When the online trajectory drifts, this distribution shift reduces predictive power.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What is the standard adaptive approach for ROMs?",{"text":79,"@type":75},"Adaptive ROMs evolve the reduced representation online using out-of-span information with nonzero residual outside the current subspace, such as full-order corrections or sensor snapshots.",{"name":81,"@type":72,"acceptedAnswer":82},"How does the proposed in-span learning mechanism differ from out-of-span adaptation?",{"text":83,"@type":75},"In-span learning streams the ROM’s own predictions, which remain inside the current reduced subspace, to reorganize the representation through incremental SVD with forgetting. 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