A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives

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DOI number:10.1137/22M1534420

Affiliation of Author(s):数学与统计学院

Journal:SIAM J. Optimization

Key Words:deep learning-based inverse problems and optimal control of PDEs

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.

Indexed by:Article

Discipline:Natural Science

First-Level Discipline:Mathematics

Document Type:J

Volume:3

Issue:3

Page Number:2314-2349

Translation or Not:no

Date of Publication:2024-06-30

Included Journals:SCI

Links to published journals:https://epubs.siam.org/doi/full/10.1137/22M1534420