Séminaire du CERAMATHS - DMATHS : exposé de Constantin Christof
Le séminaire du département de mathématiques du CERAMATHS accueillera Constantin Christof (TU Munchen, Allemagne), jeudi 25 avril 2024
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Le 25/04/2024
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14:00 - 15:00
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Séminaire
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Campus Mont Houy - Bâtiment Abel de Pujol 2 - amphi 70E
Le séminaire du département de mathématiques du CERAMATHS accueillera à 14h Constantin Christof (TU Munchen, Allemagne), jeudi 25 avril 2024, pour l'exposé suivant :
On the Identification and Optimization of Nonsmooth Superposition Operators in Semilinear Elliptic PDEs
We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc. This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation functions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero.
Responsables du séminaire :
Serge Nicaise
Bouchaïb Sodaïgui
Le séminaire du CERAMATHS - DMATHS
11.09.2023