  Nonlinear model predictive control (NMPC) often requires real-time solution
to optimization problems. However, in cases where the mathematical model is of
high dimension in the solution space, e.g. for solution of partial differential
equations (PDEs), black-box optimizers are rarely sufficient to get the
required online computational speed. In such cases one must resort to
customized solvers. This paper present a new solver for nonlinear
time-dependent PDE-constrained optimization problems. It is composed of a
sequential quadratic programming (SQP) scheme to solve the PDE-constrained
problem in an offline phase, a proper orthogonal decomposition (POD) approach
to identify a lower dimensional solution space, and a neural network (NN) for
fast online evaluations. The proposed method is showcased on a regularized
least-square optimal control problem for the viscous Burgers' equation. It is
concluded that significant online speed-up is achieved, compared to
conventional methods using SQP and finite elements, at a cost of a prolonged
offline phase and reduced accuracy.
