  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.
