  Design and optimal control problems are among the fundamental, ubiquitous
tasks we face in science and engineering. In both cases, we aim to represent
and optimize an unknown (black-box) function that associates a
performance/outcome to a set of controllable variables through an experiment.
In cases where the experimental dynamics can be described by partial
differential equations (PDEs), such problems can be mathematically translated
into PDE-constrained optimization tasks, which quickly become intractable as
the number of control variables and the cost of experiments increases. In this
work we leverage physics-informed deep operator networks (DeepONets) -- a
self-supervised framework for learning the solution operator of parametric PDEs
-- to build fast and differentiable surrogates for rapidly solving
PDE-constrained optimization problems, even in the absence of any paired
input-output training data. The effectiveness of the proposed framework will be
demonstrated across different applications involving continuous functions as
control or design variables, including time-dependent optimal control of heat
transfer, and drag minimization of obstacles in Stokes flow. In all cases, we
observe that DeepONets can minimize high-dimensional cost functionals in a
matter of seconds, yielding a significant speed up compared to traditional
adjoint PDE solvers that are typically costly and limited to relatively
low-dimensional control/design parametrizations.
