Deep Neural Networks and Finite Elements of Any Order on Arbitrary Dimensions
This provides a foundational link between deep learning and numerical methods, enabling neural networks to approximate finite element solutions broadly in computational science and engineering.
The paper tackles the problem of representing Lagrange finite element functions of any order on arbitrary-dimensional simplicial meshes using deep neural networks with ReLU and ReLU^2 activations, achieving a systematic generation of general continuous piecewise polynomial functions for the first time.
In this study, we establish that deep neural networks employing ReLU and ReLU$^2$ activation functions can effectively represent Lagrange finite element functions of any order on various simplicial meshes in arbitrary dimensions. We introduce two novel formulations for globally expressing the basis functions of Lagrange elements, tailored for both specific and arbitrary meshes. These formulations are based on a geometric decomposition of the elements, incorporating several insightful and essential properties of high-dimensional simplicial meshes, barycentric coordinate functions, and global basis functions of linear elements. This representation theory facilitates a natural approximation result for such deep neural networks. Our findings present the first demonstration of how deep neural networks can systematically generate general continuous piecewise polynomial functions on both specific or arbitrary simplicial meshes.