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 a constant volume. We successfully 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 Poisson's equation in $\mathbb{R}^2$. We use variational neural networks to approximate the solution to Poisson's equation 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. A significant advantage of this approach is its inherent parallelizability, 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：在本研究中，我们探索了使用基于神经网络的方法解决几何形状优化问题的数值解法。这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，通常需在恒定体积等几何约束下进行。我们成功开发了一种概念验证，采用灵活且可并行化的方法来处理这些问题。我们专注于一个原型问题：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。我们使用变分神经网络来近似给定区域上泊松方程的解，并通过一个神经网络表示形状，该网络近似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固有的可并行性，这使得处理参数增加变得容易。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，并扩展到伯努利型自由边界问题。解决该形状优化问题的源代码是开源且可免费获取的。