[Dottorcomp] Seminari di Matematica Applicata. Martedì 22 Marzo. Carlo Marcati.

[Stefano Lisini]

Seminari di Matematica Applicata, Dipartimento di Matematica "F. Casorati"
e Istituto del CNR IMATI "E. Magenes" di Pavia.

Martedì 22 marzo 2022, alle ore 15 precise, presso l'aula Beltrami del
Dipartimento di Matematica,

Carlo Marcati (Università di Pavia) terrà un seminario dal titolo:

Neural network and operator network approximations for elliptic PDEs

Il seminario verrà anche trasmesso in diretta su zoom al link

https://us02web.zoom.us/j/87518137128?pwd=eHg2eG9QZUdydFJEU1NESWN5a1lPQT09

ID riunione: 875 1813 7128
Passcode: 880047

Abstract.
The application of neural networks (NNs) to the numerical solution of PDEs
has seen growing popularity in the last five years: NNs have been used as
an ansatz space for the solutions, with different training approaches
(PINNs, deep Ritz methods, etc.); they have also been used to infer
discretization parameters and strategies. In this talk, I will focus on
deep ReLU NN approximation theory. I will first show how NNs accurately
approximate functions with isolated singularities, for example the
solutions to elliptic problems in polygons and polyhedra, or eigenfunctions
of problems with singular potentials that arise in quantum chemistry.  I
will then introduce operator networks, which approximate the solution
operator of PDEs.  I will, in particular, consider operator networks that,
given a fixed right-hand side, map sets of diffusion-reaction coefficients
into the space of solutions (coefficient-to-solution map).  When the
coefficients are smooth, the size of the networks can then be bounded with
respect to the H^1 norm of the error, uniformly over the parameter set.
The proofs of our approximation rates combine elliptic regularity,
classical and recent results in numerical analysis, and tools from NN
approximation theory.

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Pagina web dei Seminari di Matematica Applicata
https://matematica.unipv.it/ricerca/cicli-di-seminari/seminari-di-matematica-applicata/
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