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A First Step Towards Mesh-Free Probabilistic Shape Optimization

Stephan Schmidt, Maximilian Würschmidt

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.

A First Step Towards Mesh-Free Probabilistic Shape Optimization

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.
Paper Structure (5 sections, 25 equations, 1 figure, 3 algorithms)

This paper contains 5 sections, 25 equations, 1 figure, 3 algorithms.

Table of Contents

  1. Introduction
  2. Calculate State via PINN
  3. Calculate Shape Gradient via Probabilistic Shape Derivative
  4. Numerical Results
  5. Perspectives for a Fully Mesh-Free Probabilistic Framework

Figures (1)

  • Figure 1: Optimization results.

Theorems & Definitions (2)

  • Remark 3.1
  • Remark 3.2