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SUMMARY:Solving Roughly Forced Nonlinear PDEs via Misspecified Kernel Meth
 ods and Neural Networks - Matthieu Darcy (CALTECH (California Institute of
  Technology))
DTSTART:20250827T103000Z
DTEND:20250827T110000Z
UID:TALK234493@talks.cam.ac.uk
DESCRIPTION:Ricardo Baptista\, Edoardo Calvello\, Matthieu Darcy\, Houman 
 Owhadi\, Andrew M. Stuart\, and Xianjin Yang.\nAbstract. We consider the u
 se of Gaussian Processes (GPs) or Neural Networks (NNs) to numerically app
 rox-imate the solutions to nonlinear partial differential equations (PDEs)
  with rough forcing or sourceterms\, which commonly arise as pathwise solu
 tions to stochastic PDEs. Kernel methods have re-cently been generalized t
 o solve nonlinear PDEs by approximating their solutions as the maximuma po
 steriori estimator of GPs that are conditioned to satisfy the PDE at a fin
 ite set of collocationpoints. The convergence and error guarantees of thes
 e methods\, however\, rely on the PDE beingdefined in a classical sense an
 d its solution possessing sufficient regularity to belong to the associate
 dreproducing kernel Hilbert space. We propose a generalization of these me
 thods to handle roughlyforced nonlinear PDEs while preserving convergence 
 guarantees with an oversmoothing GP kernelthat is misspecified relative to
  the true solution&rsquo\;s regularity. This is achieved by conditioning a
 regular GP to satisfy the PDE with a modified source term in a weak sense 
 (when integrated againsta finite number of test functions). This is equiva
 lent to replacing the empirical L2-loss on the PDEconstraint by an empiric
 al negative-Sobolev norm. We further show that this loss function can beus
 ed to extend physics-informed neural networks (PINNs) to stochastic equati
 ons\, thereby resultingin a new NN-based variant termed Negative Sobolev N
 orm-PINN (NeS-PINN).
LOCATION:Seminar Room 1\, Newton Institute
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