Simultaneous and meshfree topology optimization with physics-informed Gaussian processes

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 have $\left(1\right)$ a nested nature, and $\left(2\right)$ leverage numerical solvers for design evaluations during the optimization and hence rely on discretizing the design and state 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 optimization 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 $\left(1\right)$ having a built-in continuation nature since the performance metric is optimized at the same time that the state equations are solved, and $\left(2\right)$ 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 canonical 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.