Solving PDEs on complex geometries has long become a challenge in engineering practice. The accuracy and efficiency of traditional grid-based approaches, e.g. finite elements, are strongly dependent on the quality of the underlying meshes. Therefore, much effort has been paid in developing meshless methods and methods that are less sensitive to the mesh quality. In this talk, I will introduce my previous works on finite elements and summarize some recent works on machine-learning-based meshless methods for solving elliptic PDEs on complex geometries.


In developing mesh distortion insensitive finite elements, the key is to eliminate the effect of the inverse of the Jacobian. I will introduce some new finite element formulations based on principles in mechanics, such as the principle of minimum complementary energy. In addition, using the partition-of-unity idea from mesh-less methods and fast geometric algorithms, independent finite element meshes can overlap pretty arbitrarily, which significantly reduces the effort required to mesh a complex domain and provides convenience for local refinements.


Recently, deep-learning-based methods became prevalent as alternatives to particle-based meshless methods. I will briefly introduce the main ideas and discuss their advantages and disadvantages. Finally, I will show that neural networks may not be optimal in all tasks and linear models usually do better for many smooth PDEs on irregular domains.