On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs
-
Published:2023-12-19
Issue:
Volume:
Page:
-
ISSN:1292-8119
-
Container-title:ESAIM: Control, Optimisation and Calculus of Variations
-
language:
-
Short-container-title:ESAIM: COCV
Author:
Christof Constantin,Kowalczyk Julia
Abstract
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 (PDE) 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 $H^1_{loc}(\R)$. 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. The convergence of the resulting algorithm is proven in the function space setting. 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. The paper concludes with numerical experiments which confirm the theoretical findings.
Subject
Computational Mathematics,Control and Optimization,Control and Systems Engineering
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献