Lundi 4 mars 2024 – LJLL : Carlo Marcati (Università di Pavia)

Titre : Neural Operator Surrogates for Elliptic PDEs

Résumé :

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.

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