  Inspired by applications in optimal control of semilinear elliptic partial
differential equations and physics-integrated imaging, differential equation
constrained optimization problems with constituents that are only accessible
through data-driven techniques are studied. A particular focus is on the
analysis and on numerical methods for problems with machine-learned components.
For a rather general context, an error analysis is provided, and particular
properties resulting from artificial neural network based approximations are
addressed. Moreover, for each of the two inspiring applications analytical
details are presented and numerical results are provided.
