A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives
We propose and analyze a numerical algorithm for solving a class of optimal control problems for learning-informed semilinear partial differential equations. The latter is a class of PDEs with constituents that are in principle unknown and are approximated by nonsmooth ReLU neural networks. We first...
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| Main Authors | , , |
|---|---|
| Format | Journal Article |
| Language | English |
| Published |
14.10.2022
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| Subjects | |
| Online Access | Get full text |
| DOI | 10.48550/arxiv.2210.07900 |
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| Abstract | We propose and analyze a numerical algorithm for solving a class of optimal
control problems for learning-informed semilinear partial differential
equations. The latter is a class of PDEs with constituents that are in
principle unknown and are approximated by nonsmooth ReLU neural networks. We
first show that a direct smoothing of the ReLU network with the aim to make use
of classical numerical solvers can have certain disadvantages, namely
potentially introducing multiple solutions for the corresponding state
equation. This motivates us to devise a numerical algorithm that treats
directly the nonsmooth optimal control problem, by employing a descent
algorithm inspired by a bundle-free method. Several numerical examples are
provided and the efficiency of the algorithm is shown. |
|---|---|
| AbstractList | We propose and analyze a numerical algorithm for solving a class of optimal
control problems for learning-informed semilinear partial differential
equations. The latter is a class of PDEs with constituents that are in
principle unknown and are approximated by nonsmooth ReLU neural networks. We
first show that a direct smoothing of the ReLU network with the aim to make use
of classical numerical solvers can have certain disadvantages, namely
potentially introducing multiple solutions for the corresponding state
equation. This motivates us to devise a numerical algorithm that treats
directly the nonsmooth optimal control problem, by employing a descent
algorithm inspired by a bundle-free method. Several numerical examples are
provided and the efficiency of the algorithm is shown. |
| Author | Papafitsoros, Kostas Dong, Guozhi Hintermüller, Michael |
| Author_xml | – sequence: 1 givenname: Guozhi surname: Dong fullname: Dong, Guozhi – sequence: 2 givenname: Michael surname: Hintermüller fullname: Hintermüller, Michael – sequence: 3 givenname: Kostas surname: Papafitsoros fullname: Papafitsoros, Kostas |
| BackLink | https://doi.org/10.48550/arXiv.2210.07900$$DView paper in arXiv |
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| Snippet | We propose and analyze a numerical algorithm for solving a class of optimal
control problems for learning-informed semilinear partial differential
equations.... |
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| SubjectTerms | Mathematics - Optimization and Control |
| Title | A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives |
| URI | https://arxiv.org/abs/2210.07900 |
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