Physics-informed neural network based topology optimization through continuous adjoint

In this paper, we introduce a Physics-Informed Neural Networks (PINNs)-based Topology optimization method that is free from the usual finite element analysis and is applicable for both self-adjoint and non-self-adjoint problems. This approach leverages the continuous formulation of TO along with the continuous adjoint method to obtain sensitivity. Within this approach, the Deep Energy Method (DEM)-a variant of PINN-completely supersedes traditional PDE solution procedures such as a finite-element method (FEM) based solution process. We demonstrate the efficacy of the DEM-based TO framework through three benchmark TO problems: the design of a conduction-based heat sink, a compliant displacement inverter, and a compliant gripper. The results indicate that the DEM-based TO can generate optimal designs comparable to those produced by traditional FEM-based TO methods. Notably, our DEM-based TO process does not rely on FEM discretization for either state solution or sensitivity analysis. During DEM training, we obtain spatial derivatives based on Automatic Differentiation (AD) and dynamic sampling of collocation points, as opposed to the interpolated spatial derivatives from finite element shape functions or a static collocation point set. We demonstrate that, for the DEM method, when using AD to obtain spatial derivatives, an integration point set of fixed positions causes the energy loss function to be not lower-bounded. However, using a dynamically changing integration point set can resolve this issue. Additionally, we explore the impact of incorporating Fourier Feature input embedding to enhance the accuracy of DEM-based state analysis within the TO context. The source codes related to this study are available in the GitHub repository: https://github.com/xzhao399/DEM_TO.git.