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

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.