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UID:9333@i2m.univ-amu.fr
DTSTART;TZID=Europe/Paris:20261109T140000
DTEND;TZID=Europe/Paris:20261109T150000
DTSTAMP:20261105T134804Z
URL:https://www.i2m.univ-amu.fr/evenements/tba-methodes-a-elements-finis-e
 t-de-reseaux-de-neurones-informes-par-la-physique/
SUMMARY:Nilo SCHWENCKE ( OCKHAM Team (INRIA\; LIP\, ENS Lyon\; CNRS)): Beyo
 nd PINNs: From Natural-Gradient Geometry to Hybrid Galerkin Methods
DESCRIPTION:Nilo SCHWENCKE: \nPhysics-informed neural networks and finite-e
 lement methods are commonly presented as fundamentally different paradigms
 : the former as nonlinear residual minimization\, and the latter as variat
 ional projection onto a fixed approximation space. This talk develops a co
 mmon geometric and kernel-theoretic framework for both.\n\n\n\nA parametri
 zed neural solver defines a finite-dimensional manifold in function space\
 , and natural-gradient training amounts to projecting the functional resid
 ual onto its evolving tangent space. By contrast\, a finite-dimensional Ga
 lerkin space can be viewed as a flat model manifold and\, when equipped wi
 th the energy metric\, as a reproducing-kernel Hilbert space whose kernel 
 represents a projected Green operator.\n\n\n\nThis viewpoint leads to a re
 interpretation of natural-gradient and Gauss–Newton approaches to PINNs 
 (Müller and Zeinhofer\, 2023\; Jnini\, Vella and Zeinhofer\, 2025\; Mckay
  et al.\, 2025\; Schwencke and Furtlehner\, 2025\; Schwencke et al.\, 2025
 \; Urbán\, Stefanou and Pons\, 2025\; Jnini et al.\, 2026\; Webb\, Jerad 
 and Cartis\, 2026) as Petrov–Galerkin methods for the linearized PDE res
 idual\, with test functions implicitly induced by the geometry of the neur
 al tangent space. Once the test space is treated as a design variable\, st
 rong collocation can be replaced by weak and operator-adapted tests. We ex
 ploit this freedom to construct a hybrid finite element–neural network i
 n which the finite element component enforces the equation on a stable app
 roximation space\, while the neural network is trained only in its energy-
 orthogonal complement. Numerical experiments on multiscale and nonsmooth e
 lliptic problems illustrate how this decomposition combines the stability 
 of Galerkin methods with the adaptivity of neural tangent spaces.\n\n\nThe
  talk concludes with a discussion of residual-reconstruction criteria for 
 adapting the effective tangent rank and regularizing natural-gradient trai
 ning.\n \n\n\n\nJnini\, A. et al. (2026) “Curvature-Aware Optimization 
 for High-Accuracy Physics-Informed Neural Networks.” arXiv. Available at
 : https://doi.org/10.48550/arXiv.2604.05230.\n\n\n\nJnini\, A.\, Vella\, F
 . and Zeinhofer\, M. (2025) “Gauss-newton natural gradient descent for p
 hysics-informed computational fluid dynamics\,” Computers &amp\; Fluids\
 , p. 106955.\n\n\n\nMckay\, M.B. et al. (2025) “Near-optimal Sketchy Nat
 ural Gradients for Physics-Informed Neural Networks.” Forty-second Inter
 national Conference on Machine Learning. Available at: https://openreview.
 net/forum?id=bKsZomnmqn.\n\n\n\nMüller\, J. and Zeinhofer\, M. (2023) “
 Achieving high accuracy with PINNs via energy natural gradient descent\,
 ” International Conference on Machine Learning. PMLR\, pp. 25471–25485
 . Available at: https://proceedings.mlr.press/v202/muller23b.html.\n\n\n\n
 Schwencke\, N. et al. (2025) “AMStraMGRAM: Adaptive Multi-cutoff Strateg
 y Modification for ANaGRAM.” arXiv. Available at: https://doi.org/10.485
 50/arXiv.2510.15998.\n\n\n\nSchwencke\, N. and Furtlehner\, C. (2025) “A
 NaGRAM: a natural gradient relative to adapted model for efficient PINNs l
 earning\,” The thirteenth international conference on learning represent
 ations. Available at: https://openreview.net/forum?id=o1IiiNIoaA.\n\n\n\nU
 rbán\, J.F.\, Stefanou\, P. and Pons\, J.A. (2025) “Unveiling the optim
 ization process of physics informed neural networks: How accurate and comp
 etitive can PINNs be?\,” Journal of Computational Physics\, 523\, p. 113
 656. Available at: https://doi.org/10.1016/j.jcp.2024.113656.\n\n\n\nWebb\
 , J.\, Jerad\, S. and Cartis\, C. (2026) “An Optimisation Framework for 
 the Well-Conditioned Training of Physics-Informed Neural Networks.” arXi
 v. Available at: https://doi.org/10.48550/arXiv.2607.02194.\n\n\n
CATEGORIES:Séminaire,Statistique
LOCATION:I2M Saint-Charles - Salle de séminaire\, Université Aix-Marseill
 e\, Campus Saint-Charles\, 3 Place Victor Hugo\, Marseille\, 13003\, Franc
 e
X-APPLE-STRUCTURED-LOCATION;VALUE=URI;X-ADDRESS=Université Aix-Marseille\,
  Campus Saint-Charles\, 3 Place Victor Hugo\, Marseille\, 13003\, France;X
 -APPLE-RADIUS=100;X-TITLE=I2M Saint-Charles - Salle de séminaire:geo:0,0
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