Volume-preserving geometric shape optimization of the Dirichlet energy using variational neural networks

TL;DR

The paper develops a mesh-free, parallelizable framework to minimize the Dirichlet energy over domains of fixed volume by learning both the PDE solution and a volume-preserving domain transformation. It couples a DeepRitz-style variational neural network for solving Poisson-type PDEs with SympNets that represent volume-preserving domain deformations, enabling joint optimization without explicit shape derivatives. The method handles Dirichlet and Robin boundary conditions, parametric right-hand sides, and Bernoulli-type free-boundary problems, and the authors provide open-source GeSONN code for reproduction and extension. The approach offers a scalable path toward higher-dimensional and more complex PDE shape-optimization problems by remaining mesh-free and highly parallelizable, with demonstrated accuracy against reference solutions in 2D and a clear roadmap for generalization.

Abstract

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.

Figures (18)

• Figure 1: Diagram of a fully connected neural network, where the output of each layer is a vector $z_k \in \mathbb{R}^{q_k}$, with $q_k$ the number of neurons of the layer $k$. Computing $z_k$ involves a weight matrix $W^k \in \mathcal{M}_{q_k, q_{k-1}}(\mathbb R)$, a bias vector $b^k\in \mathbb{R}^{q_k}$ and a nonlinear activation function $\sigma:\mathbb{R}\to\mathbb{R}$ applied componentwise.
• Figure 2: SympNet architecture. Here the $(T_i)_{1\leqslant i \leqslant q}$ are shear symplectic maps, while the $(\hat{\sigma}_i)_{1\leqslant i \leqslant q}$ denote their associated gradient modules.
• Figure 3: Transformation of the ball $\mathbb B^2$ by a symplectic map $\mathcal{T}$.
• Figure 4: Results of the PINN applied to the Poisson problem \ref{['eq:poisson']} with the source term \ref{['eq:parametric_source']}, on the domain given by applying the symplectic map \ref{['eq:bizaroid']} with $\lambda = 0.5$ to the annulus with inner radius $0.2$ and outer radius $1$. Top panels: results with $\mu = 0.5$; bottom panels: results with $\mu = 1.5$. Left panels: approximate solutions; right panels: errors with respect to a finite element simulation on a fine grid ($1000$ nodes per edge, around $10^6$ elements).
• Figure 5: SympNets for $\mathcal{T}_\lambda$, for $\lambda=0.5$ (upper left panel), $\lambda=1.25$ (upper right panel) and $\lambda=2$ (lower left panel). The red lines correspond to the reference shape, while the green ones correspond to the learned shape. The lower right picture is a superposition of learned shapes for $\lambda\in \{0.5, 0.875,1.25, 1.625,2\}$.

Theorems & Definitions (22)

• Remark 1: Some comments about existence and regularity of optimal shapes
• Theorem 1
• Remark 2
• Definition 1: Shear maps Arnold
• Lemma 1: JIN2020166
• Lemma 2
• proof
• Remark 3
• Remark 4
• Remark 5