[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85842-en":3,"doc-seo-85842-105":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},85842,8796095461610,"Oliver","https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c",8,"Research & Report","The Differential Neural Tangent Kernel and Its Positivity","The Neural Tangent Kernel (NTK) supports analysis of neural-network training dynamics in the over-parameterized regime, and recent advances extend NTK ideas to physics-informed neural networks (PINNs) for solving linear PDEs. For PINNs, proving NTK positivity is difficult because multiple differential operators interact. The work introduces the Differential Neural Tangent Kernel (DNTK), establishes positivity for infinite-width shallow and deep networks, and covers a broad activation class, including RePU and smooth non-polynomial functions, for all linear differential operators.","arXiv :2607 . 10200v 1 [ cs .LG] 11 Jul 2026  \nTHE DIFFERENTIAL NEURAL TANGENT KERNEL AND ITS POSITIVITY  \nBANGTI JIN AND LONGJUN WU  \nDepartment of Mathematics, The Chinese University of Hong Kong, Shatin, N. T., Hong Kong, P.R. China  \nAbstract. The Neural Tangent Kernel (NTK) is one powerful tool for analyzing the training dynamics of neural networks in the over-parameterized regime. Recently, the theoretical framework has been extended to physicsinformed neural networks (PINNs) for solving linear PDEs, one highly popular class of neural PDE solvers. In the analysis, the positivity of the associated NTK plays a fundamental role. However, establishing the positivity of the NTK for PINNs is highly challenging, due to the presence of multiple differential operators. In this work, we propose a new theoretical framework, called Differential Neural Tangent Kernel (DNTK), for analyzing PINNs through the lens of the NTK, and establish the positivity of the infinite width DNTK for both shallow and deep neural networks for a wide class of activation functions, including RePU and smooth but non-polynomial activations, for all linear differential operators. These theoretical results lay the foundation for the analysis of gradient type algorithms for training PINNs.  \nKey words: physics informed neural networks, differential neural tangent kernel, infinite-width limit, positivity, deep neural networks  \n1. Introduction  \nIn recent years, deep neural networks (DNNs) have been established as an effective tool to solve highdimensional Partial Differential Equations (PDEs) [21, 30], among which physics informed neural networks (PINNs) [39] (see [12, 32 , 33 , 41] for related contributions) represent one of the most popular classes of neural PDE solvers. Due to their ease of implementation and flexibility with various PDE models, PINNs have been widely employed for solving a wide variety of PDE problems arising in scientific and engineer disciplines, e.g. , Poisson equation with singularities [23], fluid mechanics [16, 28], solid mechanics [41], and various PDE inverse problems [7, 26 , 42], and have exhibited strong empirical performance in practice. We refer interested readers to the reviews [11, 43] for further details on the practical applications, methodological developments and numerical analysis of PINNs. Thus there is enormous interest in providing the theoretical analysis of PINNs, and important progresses have been made in terms of the analysis of the approximation error and statistical error; see, e.g.,[8, 13 , 25 , 40 , 50] for an incomplete list and the review [11] and the references therein.  \nIn practice, PINNs are often trained using gradient type algorithms, e.g., (stochastic) gradient descent, Adam [31] and L-BFGS [4], and these algorithms often perform reasonably well. Despite the impressive empirical successes, the theoretical analysis of training processes is formidably challenging. Indeed, due to the nonlinearity of the DNNs with respect to the DNN parameters, the PINN loss functions are highly nonconvex with respect to the DNN parameters and fraught with many bad local minima, and it is very challenging to prove that the chosen training algorithm converges to a global minimizer of the loss. This issue represents one outstanding challenge in the convergence analysis of PINNs. One highly powerful idea to prove the convergence of gradient type algorithm for regression tasks involving DNNs is the neural tangent kernel (NTK) framework due to [24] . In essence, in the NTK regime, the training dynamics of the DNNs is approximated by that of analytically more tractable kernel regression. The general strategy is to prove that the NTK of DNNs is positive definite at (random) initialization and remains largely constant (and thus stays positive definite) during the entire optimization process in the infinite-width case, which linearizes the DNN and allows attaining global optima [24] . In the last few years, within the NTK framewo","cbCaiiou5Zij9uBz","https://ap.wps.com/l/cbCaiiou5Zij9uBz","pdf",442381,5,1,31,"English","en",105,"# Introduction\n## Background: PINNs and NTK-based training analysis\n## Challenge: positivity of NTK for PINNs with multiple differential operators\n## Prior work and motivation for DNTK","[{\"question\":\"What problem does the paper address in PINN theory?\",\"answer\":\"It addresses the difficulty of proving positivity for the NTK associated with physics-informed neural networks, which is crucial for convergence analysis but becomes hard due to multiple differential operators.\"},{\"question\":\"What is the proposed framework called?\",\"answer\":\"The paper proposes the Differential Neural Tangent Kernel (DNTK) framework to analyze PINNs through NTK-style reasoning.\"},{\"question\":\"Under what conditions does the paper establish positivity?\",\"answer\":\"It proves positivity of the infinite-width DNTK for both shallow and deep neural networks, for a wide class of activation functions (including RePU and smooth non-polynomial activations) and for all linear differential 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problem does the paper address in PINN theory?","Question",{"text":76,"@type":77},"It addresses the difficulty of proving positivity for the NTK associated with physics-informed neural networks, which is crucial for convergence analysis but becomes hard due to multiple differential operators.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What is the proposed framework called?",{"text":81,"@type":77},"The paper proposes the Differential Neural Tangent Kernel (DNTK) framework to analyze PINNs through NTK-style reasoning.",{"name":83,"@type":74,"acceptedAnswer":84},"Under what conditions does the paper establish positivity?",{"text":85,"@type":77},"It proves positivity of the infinite-width DNTK for both shallow and deep neural networks, for a wide class of activation functions (including RePU and smooth non-polynomial activations) and for all linear differential 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