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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:r-adaptivity\, deep learning and optimal transport
- Christopher Budd (University of Bath)
DTSTART;TZID=Europe/London:20211208T170000
DTEND;TZID=Europe/London:20211208T183000
UID:TALK165148AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/165148
DESCRIPTION:PINNS (physics inspired neural networks) have rece
ntly become popular as a means of solving ODEs and
PDES by using the tools of deep learning. They ha
ve both shown promise for solving some differentia
l equations\, and have struggled to solve others.
Whilst advertised as being 'mesh free methods' the
y do rely on the use of collocation points. The ac
curacy of the numerical solution of PDEs using Fin
ite Element methods depends crucially on the choic
e of an appropriate mesh. This can be obtained an
r-adaptive strategy\, which equidistributes the er
ror over the mesh elements based on a-priori/poste
riori knowledge of the solution. The core of this
talk will describe how r-adaptivity can be useful
in the context of Deep Learning. First\, we will s
how that a one-dimensional mesh can be equidistrib
uted by training a feed forward Neural Network. Th
is approach yields better results than other stand
ard numerical methods. We will then explain the tr
aining process of Physics-informed Neural Networks
(PINNs) for solving Boundary value problems (BVPs
) and show numerical results for a reaction-diffus
ion and convection-dominated equation. It appears
that PINNs fail to be trained in the latter case u
nless the homotropy method is employed. Finally\,
we will introduce the Deep-Ritz-Network (DRN) for
solving the Poisson equation on a non-convex 2-dim
ensional domain. If the collocation points are uni
formly random sampled and fixed for the entire tra
ining process\, we obtain a solution with poor acc
uracy. On the contrary\, the adoption of an Optima
l Transport \; strategy\, which determines the
'optimal' collocation points\, results in a more
stable training process and a much more accurate s
olution
LOCATION:Seminar Room 2\, Newton Institute
CONTACT:
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