BEGIN:VCALENDAR VERSION:2.0 PRODID:-//wp-events-plugin.com//7.4.3//EN TZID:Europe/Paris X-WR-TIMEZONE:Europe/Paris BEGIN:VEVENT UID:0-333@lisn.upsaclay.fr DTSTART;TZID=Europe/Paris:20251208T140000 DTEND;TZID=Europe/Paris:20251208T140000 DTSTAMP:20251127T092410Z URL:https://www.lisn.upsaclay.fr/evenements/natural-gradients-and-kernel-m ethods-for-physics-informed-neural-networks-pinns/ SUMMARY:Natural gradients and kernel methods for Physics Informed Neural Ne tworks (PINNs) DESCRIPTION:Jury\nEmmanuel FRANK\, Chargé de recherche\, INRIA Nancy-Grand Est\, France\, Reviewer &\; Examiner\nOlga MULA\, Full Professor\, Uni versität Wien\, Austria\, Reviewer &\; Examiner (online)\nFrancis BACH \, Directeur de recherche\, INRIA Paris\, France\, Examiner\nClaire BOYER\ , Professeure des Universités\, Université Paris-Saclay\, France\, Exam iner\nVictor MICHEL-DANSAC\, Chargé de recherche\, INRIA Nancy-Grand Est\ , France\, Examiner\nCyril Furtlehner\, Chargé de recherche\, INRIA Sacl ay\, TAU team\, Supervisor\nAlena SHILOVA\, Chargé de recherche\, INRIA S aclay\, France\, Invited co-autor\nRoland MAIER\, Junior Professor\, Karls ruhe Institute of Technology\, Germany\, Invited co-autor\nAbstract\nPhysi cs-Informed Neural Networks (PINNs) have emerged in recent years as a prom ising paradigm for solving partial differential equations (PDEs) by embedd ing physical constraints directly into the training of neural networks. De spite their conceptual appeal and rapid adoption across scientific and eng ineering domains\, PINNs often suffer from limited accuracy and robustness compared to classical numerical methods. These limitations have motivated a rich body of research on algorithmic refinements\, improved training st rategies\, and theoretical analyses of their behavior. The objective of th is PhD is to advance this line of research along two complementary directi ons. On the algorithmic side\, our goal is to design more efficient and ac curate training schemes for PINNs by hybridizing tools from kernel methods and natural gradient optimization. On the theoretical side\, we aim to an chor PINNs within a rigorous mathematical framework. By situating them in the language of reproducing kernel Hilbert spaces (RKHS)\, operator theory \, and spectral analysis\, we seek to clarify their structure\, relate the m to existing approximation and variational methods\, and foster a deeper mathematical understanding of PINNs. We present our contributions across t hree papers.\nANaGRAM: We first establish a novel connection between kerne l methods and natural gradient optimization\, leading to the notion of the empirical natural gradient. Building upon this framework\, we introduce t he ANaGRAM algorithm\, a new PINN training scheme that systematically expl oits this connection. This yields improved numerical performance while als o providing a principled link between natural gradient methods and Green ’s functions.\nAMStramGRAM: Our second contribution studies the dynamics of ANaGRAM. We analyze the role of regularization by spectral cutoffs\, w hich we reinterpret in terms of enforcing isometries and relate to the the ory of Green's functions and reproducing kernels. This perspective clarifi es why cutoffs improve stability and accuracy\, and motivates a new algori thm with adaptive cutoffs\, which adjusts the degree of regularization dyn amically during training. We call this refined method AMStramGRAM.\nKernel ization of Weak Formulations: The third contribution is of a more theoreti cal nature. We show that weak and strong solutions of PDEs can both be nat urally expressed within the framework of Hilbert Riggings\, and we establi sh that these two notions in fact describe the same underlying object\, vi ewed through distinct Riggings. In particular\, we reinterpret weak formul ations as least-squares methods\, which allows us to revisit Galerkin appr oaches from a kernel-based perspective and to demonstrate how the Natural Neural Tangent Kernel (NNTK) provides a unifying framework bridging PINNs and classical variational formulations. This result highlights a deep stru ctural connection between neural PDE solvers and approximation theory.\nTa ken together\, these contributions advance both the practical efficiency a nd the theoretical foundations of PINNs. They underscore that kernel-based and geometric viewpoints are not only effective tools for improving perfo rmance\, but also provide the right language for integrating PINNs into th e broader landscape of numerical and functional analysis.\nThe online sess ion is avaible here CATEGORIES:AAC,Science des Données,Thèses et HDR LOCATION:LISN Site Plaine – Digitéo\, 1 rue René THOM 91190 Gif-sur -Yvette\, France X-APPLE-STRUCTURED-LOCATION;VALUE=URI;X-ADDRESS=1 rue René THOM 91190 Gif- sur-Yvette\, France;X-APPLE-RADIUS=100;X-TITLE=LISN Site Plaine – Digit éo:geo:0,0 END:VEVENT BEGIN:VTIMEZONE TZID:Europe/Paris X-LIC-LOCATION:Europe/Paris BEGIN:STANDARD DTSTART:20251026T020000 TZOFFSETFROM:+0200 TZOFFSETTO:+0100 TZNAME:CET END:STANDARD END:VTIMEZONE END:VCALENDAR