[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81959-en":3,"doc-seo-81959-105":31,"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":28,"seo_description":14,"update_tm":29,"read_time":30},81959,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","A Unified Perspective of Gaussian Process Approximation for Differential Equations","Gaussian processes are widely used to approximate differential equations, but the resulting numerical methods have become diverse and fragmented. A unified Bayesian perspective is presented to place these techniques into a common probabilistic framework using a derivative-matching interpretation for incorporating differential equation constraints into the likelihood. The framework simultaneously supports parameter estimation and solution approximation, and clarifies how multiple existing methods relate within one interpretation, providing consolidation and a foundation for future research.","arXiv :2607 .06292v1 [math .NA] 7 Jul 2026  \nA unified perspective of Gaussian process approximation for  \ndifferential equations  \nMengwu Guo*  \nCentrefor Mathematical Sciences, Lund University, Sweden  \nAbstract  \nThe use of Gaussian processes for approximating differential equations has expanded rapidly, leading to a growing, diverse, and fragmented body of numerical methods. We present a unified Bayesian perspective that places these techniques within a common probabilistic framework, based on a derivative matching interpretation for incorporating differential equation constraints into likelihood. This unified perspective supports both parameter estimation and solution approximation, and shows how a range of existing methods can be understood within it. This work aims to consolidate current developments and provide a foundation for future research.  \nKeywords: Gaussian process, Bayesian inference, differential equation, uncertainty quantification  \n1 Introduction  \nIn recent years, there has been a rapidly growing body of work on the use of Gaussian processes [22, 17, 14] for the approximation of ordinary and partial differential equations, particularly within the emerging field of scientific machine learning. These numerical techniques typically leverage Gaussian process priors to model solution functions, while incorporating physical constraints through differential operators. As a result, a wide range of methodologies has been developed, including various approaches to Gaussian process-based PDE approximation [23, 20, 21, 9, 12, 19, 4, 3, 16, 2], latent force models [31, 1], and Gaussian process-based approaches to parameter estimation and system identification [28, 13, 29, 15] . Despite this progress, the literature has become increasingly fragmented, with different methods often presented under varied modeling assumptions and algorithmic frameworks. Many of these techniques, however, share common underlying principles: most notably, the preservation of Gaussianity under linear operations [22], and the use of differential equation constraints in Bayesian inference [5] . This raises the question of whether these seemingly disparate methods can be understood within a single, coherent probabilistic framework.  \n* [E-mail:](E-mail: mengwu.guo@math.lu.se)[ mengwu.guo@math.lu.se](E-mail: mengwu.guo@math.lu.se).  \nThe aim of this note is to provide such a unified perspective. Building on a Bayesian formulation of Gaussian process approximation for differential equations, we show that a broad class of existing methods can be interpreted as instances ofa common probabilistic framework. At the same time, the rapid growth of methods on this theme suggests a need for consolidation. While the diversity of approaches has been valuable in investigating different modeling choices and formulations, it is increasingly important to clarify their relationships and underlying assumptions. In this spirit, this note is intended both as a technical contribution and as a pedagogical overview, aiming to synthesize existing ideas and provide a coherent foundation for future developments.  \nTo formalize the discussion, we consider differential equations of the general form:  \nL φ u (s) = gθ (u(s)) + f(s) , s ∈ D . (1)  \nHere, u denotes the (scalar-valued) solution function defined on a domain D; the operator L φ is a linear differential operator parameterized by a vector φ; the term gθ ◦ u represents a (nonlinear) source term that depends on the solution state u and is parameterized by a vector θ; finally, f is a given forcing term defined on D. This formulation covers a wide range of differential problems, including linear and nonlinear equations, and accommodates both forward and inverse problems. The objective is to infer the solution u, the parameters (φ , θ), or both, from both observational data on the solution and the governing equation.  \nThe central idea of this work is to formulate Gaussian process approximation of differential equation","cbCaiahQtzXxxQue","https://ap.wps.com/l/cbCaiahQtzXxxQue","pdf",535367,4,1,16,"English","en",105,"# Introduction\n# Unified Bayesian framework","[{\"question\":\"What problem does the paper address about Gaussian process methods for differential equations?\",\"answer\":\"It addresses the fragmentation of numerical methods that use Gaussian processes to approximate differential equations, where approaches are often presented with different assumptions and frameworks.\"},{\"question\":\"How does the paper unify different Gaussian process approximation methods?\",\"answer\":\"It proposes a unified Bayesian framework that uses a derivative-matching interpretation to encode differential equation constraints into the likelihood under a common probabilistic model.\"},{\"question\":\"What inference tasks does the unified framework support?\",\"answer\":\"The framework supports both parameter estimation and solution approximation by producing a posterior distribution that jointly characterizes the solution and unknown parameters.\"}]","A Unified Perspective of Gaussian Process Approximation for Differential Equations | 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problem does the paper address about Gaussian process methods for differential equations?","Question",{"text":76,"@type":77},"It addresses the fragmentation of numerical methods that use Gaussian processes to approximate differential equations, where approaches are often presented with different assumptions and frameworks.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How does the paper unify different Gaussian process approximation methods?",{"text":81,"@type":77},"It proposes a unified Bayesian framework that uses a derivative-matching interpretation to encode differential equation constraints into the likelihood under a common probabilistic model.",{"name":83,"@type":74,"acceptedAnswer":84},"What inference tasks does the unified framework support?",{"text":85,"@type":77},"The framework supports both parameter estimation and solution approximation by producing a posterior distribution that jointly characterizes the solution and unknown 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