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We propose and analyze a numerical algorithm for solving a class of optimalcontrol problems for learning-informed semilinear partial differentialequations. The latter is a class of PDEs with constituents that are inprinciple unknown and are approximated by nonsmooth ReLU neural networks. Wefirst show that a direct smoothing of the ReLU network with the aim to make useof classical numerical solvers can have certain disadvantages, namelypotentially introducing multiple solutions for the corresponding stateequation. This motivates us to devise a numerical algorithm that treatsdirectly the nonsmooth optimal control problem, by employing a descentalgorithm inspired by a bundle-free method. Several numerical examples areprovided and the efficiency of the algorithm is shown.

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