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SUMMARY:A Deep Ritz method with r-adaptivity for solving Partial Different
 ial Equations - David Pardo (University of the Basque Country\, BCAM - Bas
 que Center for Applied Mathematics)
DTSTART:20211118T143000Z
DTEND:20211118T150000Z
UID:TALK165448@talks.cam.ac.uk
DESCRIPTION:(joint work with Javier Omella\, Jon A. Rivera\, and Jamie M. 
 Taylor)\nThe Ritz method is a traditional method for solving symmetric and
  positive definite problems governed by Partial Differential Equations (PD
 Es). This method minimizes the energy functional\, and a first Neural Netw
 ork (NN) formulation using this method was proposed in [1].\nIn this talk\
 , we first illustrate how traditional methods for solving PDEs using NNs (
 like Deep-Ritz\, Deep Least-Squares\, and other Deep Galerkin methods) may
  suffer from strong quadrature problems\, leading to poor approximate solu
 tions. We envision four alternatives to overcome this challenge: a) Monte 
 Carlo methods\, b) adaptive integration\, c) piecewise-polynomial approxim
 ations of the NN solution\, and d) the inclusion of regularization terms i
 n the loss following the ideas of [2]. From all these methods\, we develop
  an r-adaptive method\, which falls under the category of piecewise-polyno
 mials approximations of the NN. We consider a piecewise-linear solution ov
 er a grid--allowing for exact integration--and simultaneously optimize the
  node positions (r-adaptivity) and the solution values. We show promising 
 numerical results of the r-adaptive Deep Ritz method in one- and two-dimen
 sional domains.\n\n\nWeinan E and Bing Yu\, The Deep Ritz Method: A Deep L
 earning-Based Numerical Algorithm for Solving Variational Problems. Commun
 . Math. Stat.\, vol. 6\, no. 1\, pp. 1&ndash\;12 (2018).&nbsp\;https://doi
 .org/10.1007/s40304-018-0127-z\n\n\nSiddhartha Mishra and Roberto Molinaro
 \, Estimates on the generalization error of physics informed neural networ
 ks (PINNs) for approximating PDEs. arXiv preprint arXiv:2006.16144&nbsp\; 
 (2020).\n\n
LOCATION:Seminar Room 1\, Newton Institute
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