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Author: Dong, Guozhi × Author: Hintermüller, Michael × End date: 2024 × Start date: 2020 × Type: report × Version: publishedVersion ×

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    Optimization with learning-informed differential equation constraints and its applications
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, Kostas
    Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.
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    First-order conditions for the optimal control of learning-informed nonsmooth PDEs
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2022) Dong, Guozhi; Hintermüller, Michael; Papafitsoros, Kostas; Völkner, Kathrin
    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.