WIAS Preprint No. 2940, (2022)

First-order conditions for the optimal control of learning-informed nonsmooth PDEs


  • Dong, Guozhi
    ORCID: 0000-0002-9674-6143
  • Hintermüller, Michael
    ORCID: 0000-0001-9471-2479
  • Papafitsoros, Kostas
    ORCID: 0000-0001-9691-4576
  • Völkner, Kathrin

2020 Mathematics Subject Classification

  • 49M15 65J15 65K10 35J61 68T07 49J52


  • Nonsmooth partial differential equations, data-driven models, neural networks, ReLU activation function, optimal control, PDE constrained optimization




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.

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