  In this work, we explore the numerical solution of geometric shape
optimization problems using neural network-based approaches. This involves
minimizing a numerical criterion that includes solving a partial differential
equation with respect to a domain, often under geometric constraints like
constant volume. Our goal is to develop a proof of concept using a flexible and
parallelizable methodology to tackle these problems. We focus on a prototypal
problem: minimizing the so-called Dirichlet energy with respect to the domain
under a volume constraint, involving a Poisson equation in $\mathbb R^2$. We
use physics-informed neural networks (PINN) to approximate the Poisson
equation's solution on a given domain and represent the shape through a neural
network that approximates a volume-preserving transformation from an initial
shape to an optimal one. These processes are combined in a single optimization
algorithm that minimizes the Dirichlet energy. One of the significant
advantages of this approach is its parallelizable nature, which makes it easy
to handle the addition of parameters. Additionally, it does not rely on shape
derivative or adjoint calculations. Our approach is tested on Dirichlet and
Robin boundary conditions, parametric right-hand sides, and extended to
Bernoulli-type free boundary problems. The source code for solving the shape
optimization problem is open-source and freely available.
