AONN-2: An adjoint-oriented neural network method for PDE-constrained shape optimization

TL;DR

AONN-2 is a mesh-free neural network method for PDE-constrained shape optimization that leverages shape derivatives and the DAL framework, overcoming mesh deformation issues in traditional methods.

Contribution

It extends the original AONN to shape optimization, enabling mesh-free, boundary-point-based shape updates using neural networks and shape derivatives.

Findings

Demonstrates high accuracy in shape optimization tasks.

Shows robustness across various PDE problems.

Eliminates need for mesh maintenance and correction.

Abstract

Shape optimization has been playing an important role in a large variety of engineering applications. Existing shape optimization methods are generally mesh-dependent and therefore encounter challenges due to mesh deformation. To overcome this limitation, we present a new adjoint-oriented neural network method, AONN-2, for PDE-constrained shape optimization problems. This method extends the capabilities of the original AONN method [1], which is developed for efficiently solving parametric optimal control problems. AONN-2 inherits the direct-adjoint looping (DAL) framework for computing the extremum of an objective functional and the neural network methods for solving complicated PDEs from AONN. Furthermore, AONN-2 expands the application scope to shape optimization by taking advantage of the shape derivatives to optimize the shape represented by discrete boundary points. AONN-2 is a fully mesh-free shape optimization approach, naturally sidestepping issues related to mesh deformation, with no need for maintaining mesh quality and additional mesh corrections. A series of experimental results are presented, highlighting the flexibility, robustness, and accuracy of AONN-2.