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Article overview
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First-order conditions for the optimal control of learning-informed nonsmooth PDEs | | Guozhi Dong
; Michael Hintermüller
; Kostas Papafitsoros
; Kathrin Völkner
; | | Date: |
1 Jun 2022 | | Abstract: | In this paper we study the optimal control of a class of semilinear elliptic
partial differential equations which have nonlinear constituents that are only
accessible by data and are approximated by nonsmooth ReLU neural networks. The
optimal control problem is studied in detail. In particular, the existence and
uniqueness of the state equation are shown, and continuity as well as
directional differentiability properties of the corresponding control-to-state
map are established. Based on approximation capabilities of the pertinent
networks, we address fundamental questions regarding approximating properties
of the learning-informed control-to-state map and the solution of the
corresponding optimal control problem. Finally, several stationarity conditions
are derived based on different notions of generalized differentiability. | | Source: | arXiv, 2206.00297 | | Services: | Forum | Review | PDF | Favorites | |
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