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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:A Deep Ritz method with r-adaptivity for solving P
artial Differential Equations - David Pardo (Unive
rsity of the Basque Country\, BCAM - Basque Center
for Applied Mathematics)
DTSTART;TZID=Europe/London:20211118T143000
DTEND;TZID=Europe/London:20211118T150000
UID:TALK165448AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/165448
DESCRIPTION:(joint work with Javier Omella\, Jon A. Rivera\, a
nd Jamie M. Taylor)\nThe Ritz method is a traditio
nal method for solving symmetric and positive defi
nite problems governed by Partial Differential Equ
ations (PDEs). This method minimizes the energy fu
nctional\, and a first Neural Network (NN) formula
tion using this method was proposed in [1].\nIn th
is talk\, we first illustrate how traditional meth
ods for solving PDEs using NNs (like Deep-Ritz\, D
eep Least-Squares\, and other Deep Galerkin method
s) may suffer from strong quadrature problems\, le
ading to poor approximate solutions. We envision f
our alternatives to overcome this challenge: a) Mo
nte Carlo methods\, b) adaptive integration\, c) p
iecewise-polynomial approximations of the NN solut
ion\, and d) the inclusion of regularization terms
in 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 piec
ewise-linear solution over a grid--allowing for ex
act integration--and simultaneously optimize the n
ode positions (r-adaptivity) and the solution valu
es. We show promising numerical results of the r-a
daptive Deep Ritz method in one- and two-dimension
al domains.\n\n\nWeinan E and Bing Yu\, The Deep R
itz Method: A Deep Learning-Based Numerical Algori
thm for Solving Variational Problems. Commun. Math
. Stat.\, vol. 6\, no. 1\, pp. 1&ndash\;12 (2018).
\;https://doi.org/10.1007/s40304-018-0127-z\n
\n\nSiddhartha Mishra and Roberto Molinaro\, Estim
ates on the generalization error of physics inform
ed neural networks (PINNs) for approximating PDEs.
arXiv preprint arXiv:2006.16144 \; (2020).\n\
n
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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