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.

翻译：本研究证明，采用ReLU和ReLU$^2$激活函数的深度神经网络能够有效表示任意维数多种单纯形网格上的任意阶拉格朗日有限元函数。我们针对特定网格和任意网格分别提出了两种全局表达拉格朗日单元基函数的新公式。这些公式基于单元的几何分解，融合了高维单纯形网格、重心坐标函数及线性单元全局基函数的若干深刻且必要的性质。该表示理论为这类深度神经网络提供了自然的逼近结果。我们的发现首次展示了深度神经网络如何在特定或任意单纯形网格上系统性地生成一般连续分段多项式函数。