A First Step Towards Mesh-Free Probabilistic Shape Optimization

TL;DR

This paper introduces a mesh-free probabilistic shape optimization method using neural network-based PDE solvers and Monte Carlo sampling, demonstrated on a benchmark tracking problem.

Contribution

It presents a novel probabilistic shape derivative representation evaluated via Monte Carlo, eliminating the need for mesh-based methods.

Findings

Effective shape optimization without meshing.

Neural network PDE solver on point clouds.

Successful application to a benchmark tracking problem.

Abstract

We present an initial implementation of a probabilistic PDE-constrained shape optimization algorithm. Our method is based on a novel probabilistic representation of the shape derivative, which is evaluated using Monte Carlo sampling; and does not rely on a mesh. The underlying state is represented with a neural network-based PDE solver on point clouds. The methodology is applied throughout to a benchmark tracking problem.