Machine learning aided finite element non-uniform mesh generation

We present novel deep learning methods for guiding mesh generation during finite element simulation. The finite element (FE) method is a general method for numerically solving partial differential equations. Mesh generation is an essential and critical pre-process for the finite element method. A high quality mesh ensures low time and memory cost while preserving high accuracy. However, traditional methods for guiding non-uniform mesh generation are time consuming. The state of the art requires multiple iterations of mesh generation, FE solution, and a posteriori error estimation. We propose to use deep neural networks to replace a posteriori error estimation to guide mesh refinement with reduced iterations. The training data is generated using the conventional mesh refinement methods. After supervised learning, the neural network predicts non-uniform element size distribution instantly. What we achieve is generating a high quality non-uniform mesh with little time cost.

In this thesis, we present an introduction of finite element mesh refinement and deep learning, then a review of existing deep learning methods for solving partial differential equations and generating meshes. Then we introduce our research for finite element mesh generation in several categories of PDEs, including steady 2D, 3D and transient differential equations. The proposed methods have been implemented for Poisson's equation, linear elasticity problems and the Navier-Stokes equations of fluid dynamics. In these test problems, the use of deep learning significantly reduces the time cost and produces results with accuracy close to the traditional methods. The potential and risks of the proposed methods have been discussed. The thesis ends with a conclusion and a presentation of topics that will inspire further investigation.