  Topology optimization (TO) provides a principled mathematical approach for
optimizing the performance of a structure by designing its material spatial
distribution in a pre-defined domain and subject to a set of constraints. The
majority of existing TO approaches leverage numerical solvers for design
evaluations during the optimization and hence have a nested nature and rely on
discretizing the design variables. Contrary to these approaches, herein we
develop a new class of TO methods based on the framework of Gaussian processes
(GPs) whose mean functions are parameterized via deep neural networks.
Specifically, we place GP priors on all design and state variables to represent
them via parameterized continuous functions. These GPs share a deep neural
network as their mean function but have as many independent kernels as there
are state and design variables. We estimate all the parameters of our model in
a single for loop that optimizes a penalized version of the performance metric
where the penalty terms correspond to the state equations and design
constraints. Attractive features of our approach include $(1)$ having a
built-in continuation nature since the performance metric is optimized at the
same time that the state equations are solved, and $(2)$ being
discretization-invariant and accommodating complex domains and topologies. To
test our method against conventional TO approaches implemented in commercial
software, we evaluate it on four problems involving the minimization of
dissipated power in Stokes flow. The results indicate that our approach does
not need filtering techniques, has consistent computational costs, and is
highly robust against random initializations and problem setup.
