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 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.