[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-328007-105":59,"doc-detail-328007-en":130},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","wasserstein-residuals-learning-gradient-flows-from-population-dynamics","Wasserstein Residuals: Learning Gradient Flows from Population Dynamics","","Reconstructing population dynamics from limited snapshots is tackled through Wasserstein gradient flows, where a distribution curve evolves via an energy functional. Existing Jordan–Kinderlehrer–Otto schemes require multiple optimal transport solves and become inflexible under time discretisation and large temporal gaps. A residual formulation is introduced that enforces the continuity-equation constraint exactly when the curve is a Wasserstein gradient flow, combined with data-fitting at observation times. This unifies prior approaches and yields a simulation-free stitching particle method achieving state-of-the-art benchmark results.",{"@graph":69,"@context":122},[70,84,105],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/wasserstein-residuals-learning-gradient-flows-from-population-dynamics/328007/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/wasserstein-residuals-learning-gradient-flows-from-population-dynamics/328007.png","ImageObject",300,407,{"name":92,"@type":93},"River Wang","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-31","2026-09-21",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"What problem does the paper address?","Question",{"text":112,"@type":113},"It addresses recovering the energy functional driving a population’s evolution, given only discrete snapshots over observation times.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why are JKO-based methods considered limited here?",{"text":117,"@type":113},"They rely on the Jordan–Kinderlehrer–Otto scheme, requiring costly optimal transport solves and becoming less accurate or flexible under long temporal gaps.",{"name":119,"@type":110,"acceptedAnswer":120},"What is the core idea of the proposed approach?",{"text":121,"@type":113},"It introduces a nonnegative residual loss that becomes zero exactly when the learned distribution curve is a Wasserstein gradient flow, while simultaneously fitting observed data.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},328007,1790825806,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":139,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":145,"preview_page_count":4},1099514067438,"https://ap-avatar.wpscdn.com/avatar/100002539ee87300030?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780474512215547542","arXiv :2607 .04738v1 [ stat .ML] 6 Jul 2026  \nWasserstein Residuals: Learning Gradient Flows from  \nPopulation Dynamics  \nMarkus Heinonen 1 ,2 ∗ Yair Shenfeld 1 ,3 Ricardo Baptista 1 ,4 Daniel Waxman 1 ,5 Dmitry Batenkov 1 Tim Cooijmans 1 Eli Bingham 1  \n1 Basis Research Institute 2 Aalto University 3 Brown University  \n4 University of Toronto 5 MIT  \nAbstract  \nReconstructing population dynamics is a central problem in the physical and data sciences. Often, the dynamics are modeled as a Wasserstein gradient flow (WGF):  \na curve of distributions driven by an energy functional. Though there are multiple mathematical characterizations of a WGF, the dominant algorithmic approach relies on the Jordan–Kinderlehrer–Otto (JKO) scheme. JKO-based methods are inflexible to time discretisation and require solving costly optimal transport problems. We take a residual approach, enforcing the continuity equations via a non-negative loss function whose minimum is the WGF. Combined with a data-fitting divergence, this gives a single global objective. This perspective unifies several existing methods and leads to a new particle-based method, stitching, that is simulation-free and robust to large gaps between observations. We demonstrate that the stitching method achieves state-of-the-art performance across trajectory inference benchmarks. For code [see](see github.com/BasisResearch/wasserstein-residuals)[ github.com/BasisResearch/wasserstein-residuals](see github.com/BasisResearch/wasserstein-residuals).  \n1 Introduction  \nReconstructing how a population evolves from a handful of snapshots is a central challenge across many scientific fields, ranging from computational biology (Schiebinger et al., 2019) to crowd dynamics (Maury et al., 2010) . A common language for such problems is that of Wasserstein gradient flows: curves of probability measures t →7 ρt driven by the steepest descent of an energy functional F (Ambrosio et al., 2005 ; Santambrogio, 2015) . We study the inverse problem: recovering the energy F from observed snapshots so that its gradient flow ρ := (ρt) fits the snapshots {qt } for a collection of observation times t ∈ Tobs.  \nThe dominant line of work (Bunne et al., 2022 ; Terpin et al., 2024 ; Persiianov et al., 2026) on this problem learns the functional F by relying on the Jordan–Kinderlehrer–Otto (JKO) definition of Wasserstein gradient flows (Jordan et al., 1998) . This requires solving multiple optimal transport (OT) problems, which can be costly, and lead to inaccuracies under long temporal gaps; see Figure 1 .  \nWe take a different perspective: a residual loss that vanishes exactly when ρ is a gradient flow of F. Combined with a data-fitting divergence at observed times, the objective directly enforces both the gradient flow constraint and the data fit. The perspective unifies existing paradigms—Path-Finding (Liu and Zhou, 2026) and Action Matching (Neklyudov et al., 2023)—as instantiations of one residual framework. The residual framework leads us to a new particle-based method which achieves state-of-the-art results on real-world datasets.  \nContributions. We summarize our contributions along the following three axes:  \n∗[Correspondence to](Correspondence to markus@basis.ai)[ markus@basis.ai](Correspondence to markus@basis.ai)  \n[Preprint.](Preprint.)  \nFigure 1: Stitching recovers curvature that JKO chord predictors miss. Sparse SDE snapshots (N = 20) on a sinusoidal valley at t ∈ {0, 5 , 10 , 20 , 30} . (a) True potential. (b) Stitching’s trajectory tracks the valley. (c) Lightspeed (Terpin et al., 2024) does not capture the curved trajectories of the particles; see Section 5.1 .  \n• We recast the problem as minimizing a nonnegative residual term, a framework subsuming Path-Finding (Liu and Zhou, 2026) and Action Matching (Neklyudov et al., 2023) .  \n• We introduce stitching, a simulation-free KDE-based method, which promotes the curve ρ to a first-class learnable variable alongside F and tolerates large gaps betw","cbCaidBQYyShqz9E","https://ap.wps.com/l/cbCaidBQYyShqz9E","pdf",6455516,29,"English","# Abstract\n# Introduction\n# Contributions\n# Outline\n# Background\n# Wasserstein residuals","[{\"question\":\"What problem does the paper address?\",\"answer\":\"It addresses recovering the energy functional driving a population’s evolution, given only discrete snapshots over observation times.\"},{\"question\":\"Why are JKO-based methods considered limited here?\",\"answer\":\"They rely on the Jordan–Kinderlehrer–Otto scheme, requiring costly optimal transport solves and becoming less accurate or flexible under long temporal gaps.\"},{\"question\":\"What is the core idea of the proposed approach?\",\"answer\":\"It introduces a nonnegative residual loss that becomes zero exactly when the learned distribution curve is a Wasserstein gradient flow, while simultaneously fitting observed data.\"}]","Wasserstein Residuals: Learning Gradient Flows from Population Dynamics | PDF",1789978480,73]