Neural Operator Surrogates for Elliptic PDEs

Approximating solution sets of PDEs is an important task for many
applications such as model order reduction and uncertainty
quantification. In this talk, we consider the problem of approximating
the solution operator of a PDE, viewed as a (nonlinear) map between
subsets of infinite dimensional spaces. For example, one can consider
the map between a diffusion coefficient and the solution of the Darcy
equation with this coefficient. In recent years, several techniques
based on neural networks (NN) have been developed to tackle this
problem. 

In this talk, I will present some theoretical results on the
approximation of solution operators of linear elliptic PDEs by
surrogates with the operator network architecture. I will discuss the
convergence rates of neural operators and how they depend on the
smoothness of the coefficients in the input sets. I will also outline
some recent results on surrogates for PDEs with lognormal coefficients,
based on novel sparsity results of the solutions.
~~~~~~~~~~~~~~~~

Cette présentation pourra également être suivie en ligne à partir de 
l'adresse :

https://webinaire.numerique.gouv.fr//meeting/signin/1768/creator/1434/hash/70b26c7df84aee63530924b59446fa3ca809f352