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.

翻译：本文探讨了利用基于神经网络的方法对几何形状优化问题进行数值求解。该方法涉及最小化一个数值准则，该准则包含求解关于区域的偏微分方程，且通常需满足如恒定体积等几何约束。我们的目标是开发一种概念验证，采用灵活且可并行化的方法论来处理此类问题。我们聚焦于一个原型问题：在体积约束下，关于区域最小化所谓的狄利克雷能量，该问题涉及 $\mathbb R^2$ 中的泊松方程。我们使用物理信息神经网络（PINN）来近似给定区域上泊松方程的解，并通过一个神经网络来表示形状，该网络近似于从初始形状到最优形状的保体积变换。这些过程被整合到一个单一的最小化狄利克雷能量的优化算法中。该方法的一个显著优势是其可并行化的特性，这使得添加参数易于处理。此外，它不依赖于形状导数或伴随计算。我们的方法在狄利克雷和罗宾边界条件、参数化右侧项上进行了测试，并扩展至伯努利型自由边界问题。求解该形状优化问题的源代码是开源且可自由获取的。