[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83651-en":3,"doc-seo-83651-105":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},83651,3848291630094,"Emma Wilson","https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45",8,"Research & Report","PDE-Based Framework for Generative Modeling Beyond Classical Score-Based Diffusion","A nonlinear modification of classical Ornstein–Uhlenbeck dynamics underpins an alternative generative framework for diffusion-based modeling. The dynamics admit both an interacting particle formulation and a mean-field limit described by a nonlinear Fokker–Planck equation with superlinear drift. With appropriate parameters and sufficiently large initial mass, forward evolution exhibits condensation, proven by finite-time loss of L2 regularity. A stabilized reverse-time PDE reconstructs the initial distribution from the asymptotic forward state, extending score-based paradigms. Numerical schemes for forward and reverse processes match continuous asymptotics and enable density filtering via iterative sampling in 1D and 2D.","2 Jul 2026  \nA PDE-Based Framework for Generative Modeling Beyond Classical  \nScore-Based Diffusion  \nMichael Herty  \nInstitute for Geometry and Practical Mathematics  \nRWTH Aachen University, Germany  \nExtraordinary Professor, Department of Mathematics and Applied Mathematics University of Pretoria, South Africa  \n[herty@igpm.rwth-aachen.de](herty@igpm.rwth-aachen.de)  \nHoracio Tettamanti  \nDepartment of Mathematics “F. Casorati”  \nUniversity of Pavia, Italy  \nhoracio.tettamanti01@universitadipavia.it  \nJuly 3, 2026  \n[math .NA]  \nAbstract  \nWe introduce an alternative generative framework based on a nonlinear modification of the classical Ornstein– Uhlenbeck dynamics. The proposed dynamics admits both a microscopic description through an interacting particle system and, in the mean-field limit, a macroscopic formulation given by a nonlinear Fokker–Planck equation with a superlinear drift term. We show that, for suitable choices of the model parameters and sufficiently large initial mass, the forward dynamics exhibits condensation phenomena by proving the loss of L2 regularity of  \narXiv :2607 .02349v1  \nthe solution in finite time. Building upon this formulation, we derive a stabilized reverse-time partial differential equation that reconstructs the initial distribution from the asymptotic state of the forward dynamics, thereby extending the generative paradigm beyond the classical score-based framework. Furthermore, we introduce numerical discretizations of both the forward and reverse processes that accurately capture the asymptotic behavior of the continuous model while successfully reconstructing the initial distribution. Numerical experiments in one and two spatial dimensions validate the proposed methodology and illustrate its application to density filtering through successive iterations of the generative process.  \n1 Introduction  \nIn recent years, generative diffusion models have emerged as a powerful class of generative models, demonstrating remarkable capabilities across various domains such as computer vision, natural language processing, multi-modal modeling, among many others [4, 14 , 20 , 22 , 26 , 28 , 37] . Several approaches have been developed, with score-based generative models [36] and denoising diffusion probabilistic models [21] emerging as two of the most influential methodologies. We do not aim to review all existing work on diffusion models but follow the view point of [10] and we refer to this publication for a more detailed referee of existing approaches. Furthermore, we point the interest reader to following references for more detailed review and novel mathematical approaches recently developed [12, 29 , 35 , 32 , 33 , 38 , 41] .  \nIn general, we may state, the goal of a generative model is to produce new samples that are statistically consistent with a given dataset. Formally, let {xi} be samples drawn from an underlying distribution f0 ∈ P (Rd ), where d denotes the dimension of the ambient space. The objective is to generate a new set of samples {x˜i} that follow the same (or a similar) underlying distribution. A classical approach to nonparametric estimation of the underlying distribution is kernel density estimation, which builds an approximation of f0 from the N available samples [2, 18 , 34] . While effective in low-dimensional settings, this method suffers from the curse of dimensionality, limiting its applicability in high dimensions. To overcome this curse of dimensionality, generative models have emerged as a powerful class of methods for learning and sampling from high-dimensional probability distributions.  \nThe general idea underlying these models consists of two complementary stages: a forward process and a backward process. During the forward step, the original data are progressively corrupted by noise, transforming the underlying distribution into a simpler reference distribution, typically chosen to be Gaussian. The backward step aims at reversing this transformation, thereby re","cbCaij1Y9VT68BiR","https://ap.wps.com/l/cbCaij1Y9VT68BiR","pdf",6887628,6,1,22,"English","en",105,"# Abstract\n# Introduction\n## Generative diffusion models and motivation\n## Forward/backward process and score functions\n## PDE perspective and Ornstein–Uhlenbeck baseline","[{\"question\":\"What is the core idea of the proposed generative framework?\",\"answer\":\"It replaces the classical Ornstein–Uhlenbeck dynamics with a nonlinear modification that admits both a particle-level description and a mean-field PDE formulation for generation and reconstruction.\"},{\"question\":\"How does the forward process behave and why is condensation important?\",\"answer\":\"Under suitable parameters and sufficiently large initial mass, the forward dynamics develop condensation, established through finite-time loss of L2 regularity, indicating a strong structural change in the solution.\"},{\"question\":\"How is the initial distribution reconstructed in this approach?\",\"answer\":\"A stabilized reverse-time partial differential equation reconstructs the initial distribution from the asymptotic state produced by the forward dynamics, thereby extending beyond the classical score-based framework.\"}]",1784189521,55,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"pde-based-framework-for-generative-modeling-beyond-classical-score-based-diffusion","",{"@graph":36,"@context":86},[37,54,69],{"@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":53},"https://docshare.wps.com/document/pde-based-framework-for-generative-modeling-beyond-classical-score-based-diffusion/83651/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-27","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What is the core idea of the proposed generative framework?","Question",{"text":76,"@type":77},"It replaces the classical Ornstein–Uhlenbeck dynamics with a nonlinear modification that admits both a particle-level description and a mean-field PDE formulation for generation and reconstruction.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the forward process behave and why is condensation important?",{"text":81,"@type":77},"Under suitable parameters and sufficiently large initial mass, the forward dynamics develop condensation, established through finite-time loss of L2 regularity, indicating a strong structural change in the solution.",{"name":83,"@type":74,"acceptedAnswer":84},"How is the initial distribution reconstructed in this approach?",{"text":85,"@type":77},"A stabilized reverse-time partial differential equation reconstructs the initial distribution from the asymptotic state produced by the forward dynamics, thereby extending beyond the classical score-based 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